toric topology - arxiv.org e-print archivetoric topology victor m. buchstaber taras e. panov steklov...

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Toric Topology

Victor M. Buchstaber

Taras E. Panov

Steklov Mathematical Institute, Russian Academy of Sciences,Gubkina Street 8, 119991 Moscow, Russia

E-mail address : [email protected]

Department of Mathematics and Mechanics, Moscow State Univer-sity, Leninskie Gory, 119991 Moscow, Russia

Institute for Information Transmission Problems, Russian Acad-emy of Sciences, Moscow

Institute of Theoretical and Experimental Physics, MoscowE-mail address : [email protected]

http://arxiv.org/abs/1210.2368v3

Abstract. Toric topology emerged in the end of the 1990s on the bordersof equivariant topology, algebraic and symplectic geometry, combinatorics andcommutative algebra. It has quickly grown up into a very active area withmany interdisciplinary links and applications, and continues to attract expertsfrom different fields.

The key players in toric topology are moment-angle manifolds, a family ofmanifolds with torus actions defined in combinatorial terms. Their construc-tion links to combinatorial geometry and algebraic geometry of toric varietiesvia the related notion of a quasitoric manifold. Discovery of remarkable geo-metric structures on moment-angle manifolds led to seminal connections withthe classical and modern areas of symplectic, Lagrangian and non-Kahler com-plex geometry. A related categorical construction of moment-angle complexesand their generalisations, polyhedral products, provides a universal framework

for many fundamental constructions of homotopical topology. The study ofpolyhedral products is now evolving into a separate area of homotopy theory,with strong links to other areas of toric topology. A new perspective on torusaction has also contributed to the development of classical areas of algebraictopology, such as complex cobordism.

The book contains lots of open problems and is addressed to expertsinterested in new ideas linking all the subjects involved, as well as to graduatestudents and young researchers ready to enter into a beautiful new area.

Contents

Introduction 1Chapter guide 2Acknowledgments 5

Chapter 1. Geometry and combinatorics of polytopes 71.1. Convex polytopes 71.2. Gale duality and Gale diagrams 151.3. Face vectors and Dehn–Sommerville relations 201.4. Characterising the face vectors of polytopes 25

Polytopes: additional topics 311.5. Nestohedra and graph-associahedra 311.6. Flagtopes and truncated cubes 441.7. Differential algebra of combinatorial polytopes 501.8. Families of polytopes and differential equations 55

Chapter 2. Combinatorial structures 612.1. Polyhedral fans 622.2. Simplicial complexes 652.3. Barycentric subdivision and flag complexes 702.4. Alexander duality 732.5. Classes of triangulated spheres 762.6. Triangulated manifolds 812.7. Stellar subdivisions and bistellar moves 842.8. Simplicial posets and simplicial cell complexes 872.9. Cubical complexes 89

Chapter 3. Combinatorial algebra of face rings 973.1. Face rings of simplicial complexes 983.2. Tor-algebras and Betti numbers 1033.3. Cohen–Macaulay complexes 1113.4. Gorenstein complexes and Dehn–Sommerville relations 1153.5. Face rings of simplicial posets 118

Face rings: additional topics 1273.6. Cohen–Macaulay simplicial posets 1273.7. Gorenstein simplicial posets 1313.8. Generalised Dehn–Sommerville relations 133

Chapter 4. Moment-angle complexes 1354.1. Basic definitions 137

iii

iv CONTENTS

4.2. Polyhedral products 1414.3. Homotopical properties 1454.4. Cell decomposition 1494.5. Cohomology ring 1504.6. Bigraded Betti numbers 1554.7. Coordinate subspace arrangements 162

Moment-angle complexes: additional topics 1694.8. Free and almost free torus actions on moment-angle complexes 1694.9. Massey products in the cohomology of moment-angle complexes 1754.10. Moment-angle complexes from simplicial posets 178

Chapter 5. Toric varieties and manifolds 1855.1. Classical construction from rational fans 1855.2. Projective toric varieties and polytopes 1895.3. Cohomology of toric manifolds 1915.4. Algebraic quotient construction 1945.5. Hamiltonian actions and symplectic reduction 201

Chapter 6. Geometric structures on moment-angle manifolds 2076.1. Intersections of quadrics 2076.2. Moment-angle manifolds from polytopes 2106.3. Symplectic reduction and moment maps revisited 2156.4. Complex structures on intersections of quadrics 2186.5. Moment-angle manifolds from simplicial fans 2216.6. Complex structures on moment-angle manifolds 2256.7. Holomorphic principal bundles and Dolbeault cohomology 2296.8. Hamiltonian-minimal Lagrangian submanifolds 236

Chapter 7. Half-dimensional torus actions 2457.1. Locally standard actions and manifolds with corners 2467.2. Toric manifolds and their quotients 2487.3. Quasitoric manifolds 2497.4. Locally standard T -manifolds and torus manifolds 2637.5. Topological toric manifolds 2847.6. Relationship between different classes of T -manifolds 2887.7. Bounded flag manifolds 2907.8. Bott towers 2937.9. Weight graphs 308

Chapter 8. Homotopy theory of polyhedral products 3218.1. Rational homotopy theory of polyhedral products 3228.2. Wedges of spheres and connected sums of sphere products 3288.3. Stable decompositions of polyhedral products 3338.4. Loop spaces, Whitehead and Samelson products 3388.5. The case of flag complexes 347

Chapter 9. Torus actions and complex cobordism 3559.1. Toric and quasitoric representatives in complex bordism classes 3559.2. The universal toric genus 365

CONTENTS v

9.3. Equivariant genera, rigidity and fibre multiplicativity 3729.4. Isolated fixed points: localisation formulae 3759.5. Quasitoric manifolds and genera 382

Appendix A. Commutative and homological algebra 387A.1. Algebras and modules 387A.2. Homological theory of graded rings and modules 390A.3. Regular sequences and Cohen–Macaulay algebras 397A.4. Formality and Massey products 402

Appendix B. Algebraic topology 407B.1. Homotopy and homology 407B.2. Elements of rational homotopy theory 418B.3. Group actions and equivariant cohomology 420B.4. Eilenberg–Moore spectral sequences 424

Appendix C. Categorical constructions 427C.1. Diagrams and model categories 427C.2. Algebraic model categories 432C.3. Homotopy limits and colimits 438

Appendix D. Bordism and cobordism 441D.1. Bordism of manifolds 441D.2. Thom spaces and cobordism functors 442D.3. Oriented and complex bordism 445D.4. Structure results 452D.5. Ring generators 453D.6. Invariant stably complex structures 456

Appendix E. Formal group laws and Hirzebruch genera 459E.1. Elements of the theory of formal group laws 459E.2. Formal group law of geometric cobordisms 460E.3. Hirzebruch genera 463

Bibliography 469

Index 483

Introduction

Traditionally, the study of torus actions on topological spaces has been consid-ered as a classical field of algebraic topology. Specific problems connected with torusactions arise in different areas of mathematics and mathematical physics, which re-sults in permanent interest in the theory, constant source of new applications andpenetration of new ideas into topology.

Since the 1970s, algebraic and symplectic viewpoints on torus actions haveenriched the subject with new combinatorial ideas and methods, largely based onthe convex-geometric concept of polytopes.

The study of algebraic torus actions on algebraic varieties has quickly developedinto a much successful branch of algebraic geometry, known as toric geometry. Itgives a bijection between, on the one hand, toric varieties, which are complexalgebraic varieties equipped with an action of an algebraic torus with a dense orbit,and on the other hand, fans, which are combinatorial objects. The fan allowsone to completely translate various algebraic-geometric notions into combinatorics.Projective toric varieties correspond to fans which arise from convex polytopes. Avaluable aspect of this theory is that it provides many explicit examples of algebraicvarieties, leading to applications in deep subjects such as the singularity theory andmirror symmetry.

In symplectic geometry, since the early 1980s there has been much activity inthe field of Hamiltonian group actions on symplectic manifolds. Such an actiondefines the moment map from the manifold to a Euclidean space (more precisely,the dual Lie algebra of the torus) whose image is a convex polytope. If the torus hashalf the dimension of the manifold the moment map image determines the manifoldup to equivariant symplectomorphism. The class of polytopes which can arise asthe images of moment maps can be described explicitly, together with an effectiveprocedure of recovering a symplectic manifold from such a polytope. In symplecticgeometry, as in algebraic geometry, one translates various geometric constructionsinto the language of convex polytopes and combinatorics.

There is a tight relationship between the algebraic and the symplectic pictures:a projective embedding of a toric manifold determines a symplectic form and amoment map. The image of the moment map is a convex polytope that is dual tothe fan. In both the smooth algebraic-geometric and the symplectic situations, thecompact torus action is locally isomorphic to the standard action of (S1)n on Cn

by rotation of the coordinates. Thus the quotient of the manifold by this actionis naturally a manifold with corners, stratified according to the dimension of thestabilisers, and each stratum can be equipped with data that encodes the isotropytorus action along that stratum. Not only does this structure of the quotient providea powerful means of investigating the action, but some of its subtler combinatorialproperties may also be illuminated by a careful study of the equivariant topology

1

2 INTRODUCTION

of the manifold. Thus, it should come as no surprise that since the beginning of the1990s, the ideas and methodology of toric varieties and Hamiltonian torus actionshave started penetrating back into algebraic topology.

By 2000, several constructions of topological analogues of toric varieties andsymplectic toric manifolds have appeared in the literature, together with differentseemingly unrelated realisations of what later has become known as the moment-angle manifolds. We tried to systematise both known and emerging links betweentorus actions and combinatorics in our 2000 paper [54] in Russian MathematicalSurveys, where the terms ‘moment-angle manifold’ and ‘moment-angle complex’first appeared. Two years later it grew up into a book ‘Torus Actions and TheirApplications in Topology and Combinatorics’ [55] published by the AMS in 2002(the extended Russian edition [57] appeared in 2004). The title ‘Toric Topology’coined by our colleague Nigel Ray became official after the 2006 Osaka conferenceunder the same name. Its proceedings volume [150] contained many importantcontributions to the subject, as well as the introductory survey ‘An invitation totoric topology: vertex four of a remarkable tetrahedron’ by Buchstaber and Ray.The vertices of the ‘toric tetrahedron’ are topology, combinatorics, algebraic andsymplectic geometry, and it symbolised much strengthened links between thesesubjects. With many young researchers entering the subject and conferences heldaround the world every year, toric topology has definitely grown up into a maturearea. Its various aspects are presented in this monograph, with an intention toconsolidate the foundations and stimulate further applications.

Chapter guide

1

Polytopes

2

Combinatorialstructures

3

Facerings

4

Moment-anglecomplexes

❩❩❩❩❩⑦

5

Toricvarieties

6

Moment-anglemanifolds

7

Half-dimtorus actions

8

Homotopytheory

9

Cobordism

At the heart of toric topology lies a class of torus actions whose orbit spacesare highly structured in combinatorial terms, that is, have lots of orbit types tiedtogether in a nice combinatorial way. We use the generic terms toric space and toricobject to refer to a topological space with a nice torus action, or to a space produced

CHAPTER GUIDE 3

from a torus action via different standard topological or categorical constructions.Examples of toric spaces include toric varieties, toric and quasitoric manifolds andtheir generalisations, moment-angle manifolds, moment-angle complexes and theirBorel constructions, polyhedral products, complements of coordinate subspace ar-rangements, intersections of real or Hermitian quadrics, etc.

Each chapter and most sections have their own introductions with more specificinformation about the contents.

In Chapter 1 we collect background material related to convex polytopes, in-cluding basic convex-geometric constructions and the combinatorial theory of facevectors. The famous g-theorem describing integer sequences that can be the facevectors of simple (or simplicial) polytopes was one of the most striking applicationsof toric geometry to combinatorics. The concept of Gale duality and Gale diagramsare important tools for the study of moment-angle manifolds via intersections ofquadrics. In the additional sections we describe several combinatorial constructionsproviding families of simple polytopes, including nestohedra, graph associahedra,flagtopes and truncated cubes. The classical series of permutahedra and associahe-dra (Stasheff polytopes) are particular examples. The construction of nestohedratakes its origin in singularity and representation theory. We develop a differentialalgebraic formalism which links the generating series of nestohedra to classical par-tial differential equations. The potential of truncated cubes in toric topology is yetto be fully exploited, as they provide an immense source of explicitly constructedtoric spaces.

In Chapter 2 we describe systematically combinatorial structures that appearin the orbit spaces of toric objects. Besides convex polytopes, these include fans,simplicial and cubical complexes, simplicial posets. All these structures are objectsof independent interest for combinatorialists, and we emphasised the aspects oftheir combinatorial theory most relevant to subsequent topological applications.

The subject of Chapter 3 is the algebraic theory of face rings (also known asStanley–Reisner rings) of simplicial complexes, and their generalisations to simpli-cial posets. With the appearance of the face rings in the beginning of the 1970s inthe work of Reisner and Stanley many combinatorial problems were translated intothe language of commutative algebra, which paved the way for their solution usingthe extensive machinery of algebraic and homological methods. Algebraic tools usedfor attacking combinatorial problems included regular sequences, Cohen–Macaulayand Gorenstein rings, Tor-algebras, local cohomology, etc. A whole new thrivingfield appeared on the borders of combinatorics and algebra, which has since becomeknown as combinatorial commutative algebra.

Chapter 4 is the first ‘toric’ chapter of the book; it links the combinatorialand algebraic constructions of the previous chapters to the world of toric spaces.The concept of the moment-angle complex ZK is introduced as a functor from thecategory of simplicial complexes K to the category of topological spaces with torusactions and equivariant maps. When K is a triangulated manifold, the moment-angle complex ZK consists a free orbit Z∅ consisting of singular points. Removingthis orbit we obtain an open manifold ZK\Z∅, which satisfies the relative version ofPoincare duality. Combinatorial invariants of simplicial complexes K therefore canbe described in terms of topological characteristics of the corresponding moment-angle complexes ZK. In particular, the face numbers of K, as well as the moresubtle bigraded Betti numbers of the face ring Z[K] can be expressed in terms of

4 INTRODUCTION

the cellular cohomology groups of ZK. The integral cohomology ring H∗(ZK) isshown to be isomorphic to the Tor-algebra TorZ[v1,...,vm](Z[K],Z). The proof buildsupon a construction of a ring model for cellular cochains of ZK and the corre-sponding cellular diagonal approximation, which is functorial with respect to mapsof moment-angle complexes induced by simplicial maps of K. This functorial prop-erty of the cellular diagonal approximation for ZK is quite special, due to the lack ofsuch a construction for general cell complexes. Another result of Chapter 4 is a ho-motopy equivalence (an equivariant deformation retraction) from the complementU(K) of the arrangement of coordinate subspaces in Cm determined by K to themoment-angle complex ZK. Particular cases of this result are known in toric geom-etry and geometric invariant theory. It opens a new perspective on moment-anglecomplexes, linking them to the theory of configuration spaces and arrangements.

We tried to make the material of the first four chapters of the book accessiblefor an undergraduate student, or a reader with a very basis knowledge of algebraand topology. The general algebraic and topological constructions required hereare collected in Appendices A and B respectively. More experienced readers mayrefer to these appendices purely for terminology and notation.

Toric varieties are the subject of Chapter 5. This is an extensive area with vastliterature available. We outline the influence of toric geometry on the emergenceof toric topology and emphasise combinatorial, topological and symplectic aspectsof toric varieties. The construction of moment-angle manifolds via nondegenerateintersections of Hermitian quadrics in Cm, motivated by symplectic geometry, isalso discussed here. Some basic knowledge of algebraic geometry may be requiredin Chapter 5. Appropriate references are given in the introduction to the chapter.

Geometry of moment-angle manifolds is studied in Chapter 6. The construc-tion of moment-angle manifolds as the level sets of toric moment maps is taken asthe starting point for the systematic study of intersections of Hermitian quadricsvia Gale duality. Following a remarkable discovery by Bosio and Meersseman ofcomplex-analytic structures on moment-angle manifolds corresponding to simplepolytopes, we proceed by showing that moment-angle manifolds corresponding toa more general class of complete simplicial fans can also be endowed with complex-analytic structures. The resulting family of non-Kahler complex manifolds includesthe classical series of Hopf and Calabi–Eckmann manifolds. We also describe im-portant invariants of these complex structures, such as the Hodge numbers andDolbeault cohomology rings, study holomorphic torus principal bundles over toricvarieties, and establish collapse results for the relevant spectral sequences. Weconclude by exploring the construction of A. E. Mironov providing a vast familyof Lagrangian submanifolds with special minimality properties in complex space,complex projective space and other toric varieties. Like many other geometricconstructions in this chapter, it builds upon the realisation of the moment-anglemanifold as an intersection of quadrics.

In Chapter 7 we discuss several topological constructions of even-dimensionalmanifolds with an effective action of a torus of half the dimension of the manifold.They can be viewed as topological analogues and generalisations of compact non-singular toric varieties (or toric manifolds). These include quasitoric manifolds ofDavis and Januszkiewicz, torus manifolds of Hattori and Masuda, and topologicaltoric manifolds of Ishida, Fukukawa and Masuda. For all these classes of toricobjects, the equivariant topology of the action and the combinatorics of the orbit

ACKNOWLEDGMENTS 5

spaces interact in a nice and peculiar way, leading to a host of results linkingtopology with combinatorics. We also discuss the relationship with GKM-manifolds(named after Goresky, Kottwitz and MacPherson), another class of toric objectstaking its origin in symplectic topology.

Homotopy-theoretical aspects of toric topology are the subject of Chapter 8.This is now a very active area. Homotopy techniques brought to bear on the studyof polyhedral products and other toric spaces include model categories, homotopylimits and colimits, higher Whitehead and Samelson products. The required infor-mation about categorical methods in topology is collected in Appendix C.

The final Chapter 9 we review applications of toric methods in the classicalfield of algebraic topology, complex bordism and cobordism. It is a generalisedcohomology theory that combines both geometric intuition and elaborated algebraictechniques. The toric viewpoint brings an entirely new perspective on complexbordism theory in both its non-equivariant and equivariant versions.

The later chapters require more specific knowledge of algebraic topology, suchas characteristic classes and spectral sequences, for which we recommend respec-tively the classical book of Milnor and Stasheff [227] and the excellent guide byMcCleary [215]. Basic facts and constructions from bordism and cobordism theoryare given in Appendix D, while the related techniques of formal group laws andgenera are reviewed in Appendix E.

Acknowledgments

We wish to express our deepest thanks to

· our teacher Sergei Petrovich Novikov for encouragement and support ofour research on toric topology;· our friends and coathors Mikiya Masuda and Nigel Ray for long and much

fruitful collaboration;· Peter Landweber for most helpful suggestions on improving the text;· all our colleagues who participated in conferences on toric topology for

the insight gained from discussions following talks and presentations.

The work was supported by the Russian Science Foundation (grant no. 14-11-00414). We also thank the Russian Foundation for Basic Research, the Presidentof the Russian Federation Grants Council and Dmitri Zimin’s ‘Dynasty’ foundationfor their support of our research related to this monograph.

CHAPTER 1

Geometry and combinatorics of polytopes

This chapter is an introductory survey of the geometric and combinatorialtheory of convex polytopes, with the emphasis on those of its aspects related to thetopological applications later in the book. We do not assume any specific knowledgeof the reader here. Algebraic definitions (graded rings and algebras) required in theend of this chapter are contained in Section A.1 of the Appendix.

Convex polytopes have been studied since ancient times. Nowadays both combi-natorial and geometrical aspects of polytopes are presented in numerous textbooksand monographs. Among them are the classical monograph [139] by Grunbaum andZiegler’s more recent lectures [325]. Face vectors and other combinatorial topicsare discussed in books by McMullen–Shephard [221], Brønsted [42], and the surveyarticle [187] by Klee and Kleinschmidt; while Yemelichev–Kovalev–Kravtsov [319]focus on applications to linear programming and optimisation. All these sourcesmay be recommended for the subsequent study of the theory of polytopes, andcontain a host of further references.

1.1. Convex polytopes

Definitions and basic constructions. Let Rn be n-dimensional Euclideanspace with the scalar product 〈 , 〉. There are two constructively different ways todefine a convex polytope in Rn:

Definition 1.1.1. A convex polytope is the convex hull conv(v1, . . . , vq) of afinite set of points v1, . . . , vq ∈ Rn.

Definition 1.1.2. A convex polyhedron P is a nonempty intersection of finitelymany half-spaces in some Rn:

(1.1) P =x ∈ Rn : 〈a i, x 〉+ bi > 0 for i = 1, . . . ,m

,

where a i ∈ Rn and bi ∈ R. A convex polytope is a bounded convex polyhedron.

All polytopes in this book will be convex. The two definitions above producethe same geometrical object, i.e. a subset of Rn is the convex hull of a finite pointset if and only if it is a bounded intersection of finitely many half-spaces. Thisclassical fact is proved in many textbooks on polytopes and convex geometry, andit lies at the heart of many applications of polytope theory to linear programmingand optimisation, see e.g. [325, Theorem 1.1].

The dimension of a polyhedron is the dimension of its affine hull. We oftenabbreviate a ‘polyhedron of dimension n’ to n-polyhedron. A supporting hyperplaneof P is an affine hyperplane H which has common points with P and for whichthe polyhedron is contained in one of the two closed half-spaces determined bythe hyperplane. The intersection P ∩ H with a supporting hyperplane is calleda face of the polyhedron. Denote by ∂P and intP = P \ ∂P the topological

7

8 1. GEOMETRY AND COMBINATORICS OF POLYTOPES

boundary and interior of P respectively. In the case dimP = n the boundary ∂Pis the union of all faces of P . Each face of an n-polyhedron (n-polytope) is itself apolyhedron (polytope) of dimension 6 n. Zero-dimensional faces are called vertices,one-dimensional faces are edges, and faces of codimension one are facets .

Two polytopes P ⊂ Rn1 and Q ⊂ Rn2 of the same dimension are said tobe affinely equivalent (or affinely isomorphic) if there is an affine map Rn1 → Rn2

establishing a bijection between the points of the two polytopes. Two polytopes arecombinatorially equivalent if there is a bijection between their faces preserving theinclusion relation. Note that two affinely isomorphic polytopes are combinatoriallyequivalent, but the opposite is not true.

The faces of a given polytope P form a partially ordered set (a poset) withrespect to inclusion. It is called the face poset of P . Two polytopes are combina-torially equivalent if and only if their face posets are isomorphic.

Definition 1.1.3. A combinatorial polytope is a class of combinatorially equiv-alent polytopes.

Many topological constructions later in this book will depend only on the com-binatorial equivalence class of a polytope. Nevertheless, it is always helpful, andsometimes necessary, to keep in mind a particular geometric representative P ratherthan thinking in terms of abstract posets. Depending on the context, we shall de-note by P , Q, etc., geometric polytopes or their combinatorial equivalent classes(combinatorial polytopes). Whenever we consider both geometric and combinato-rial polytopes, we shall use the notation P ≈ Q for combinatorial equivalence.

We refer to (1.1) as a presentation of the polyhedron P by inequalities. Theseinequalities contain more information than the polyhedron P , for the followingreason. It may happen that some of the inequalities 〈a i, x〉 + bi > 0 can be re-moved from the presentation without changing P ; we refer to such inequalities asredundant. A presentation without redundant inequalities is called irredundant. Anirredundant presentation of a given polyhedron is unique up to multiplication ofpairs (a i, bi) by positive numbers.

Example 1.1.4 (simplex and cube). An n-dimensional simplex ∆n is the con-vex hull of n + 1 points in Rn that do not lie on a common affine hyperplane.All faces of an n-simplex are simplices of dimension 6 n. Any two n-simplicesare affinely equivalent. Let e1, . . . , en be the standard basis in Rn. The n-simplexconv(0, e1, . . . , en) is called standard. Equivalently, the standard n-simplex is spec-ified by the n+ 1 inequalities

(1.2) xi > 0 for i = 1, . . . , n, and − x1 − · · · − xn + 1 > 0.

The regular n-simplex is the convex hull of the endpoints of e1, . . . , en+1 in Rn+1.The standard n-cube is given by

(1.3) In = [0, 1]n = (x1, . . . , xn) ∈ Rn : 0 6 xi 6 1 for i = 1, . . . , n.Equivalently, the standard n-cube is the convex hull of 2n points (ε1, . . . , εn) ∈ Rn,where εi = 0 or 1.Whenever we work with combinatorial polytopes, we shall referto any polytope combinatorially equivalent to In as a cube, and denote it by In.

The cube In has 2n facets. We denote by F 0k the facet specified by the equation

xk = 0, and by F 1k that specified by the equation xk = 1, for 1 6 k 6 n.

1.1. CONVEX POLYTOPES 9

Simple and simplicial polytopes. Polarity. The notion of a generic poly-tope depends on the choice of definition. Below we describe the two possibilities.

A set of q > n points in Rn is in general position if no n + 1 of them lie ona common affine hyperplane. Now, assuming Definition 1.1.1, we may say that apolytope is generic if it is the convex hull of a set of generally positioned points.This implies that all faces of the polytope are simplices, i.e. every facet has theminimal number of vertices (namely, n). Polytopes with this property are calledsimplicial .

Assuming Definition 1.1.2, a presentation (1.1) is said to be generic if P isnonempty and the hyperplanes defined by the equations 〈a i, x 〉 + bi = 0 are ingeneral position at any point of P (that is, for any x ∈ P the normal vectors a iof the hyperplanes containing x are linearly independent). If presentation (1.1) isgeneric, then P is n-dimensional. If P is a polytope, then the existence of a genericpresentation implies that P is simple, that is, exactly n facets meet at each vertexof P . Each face of a simple polytope is again a simple polytope. Every vertex ofa simple polytope has a neighbourhood affinely equivalent to a neighbourhood of0 in the positive orthant Rn>. It follows that every vertex is contained in exactly nedges, and each subset of k edges with a common vertex spans a k-face.

A generic presentation may contain redundant inequalities, but, for any suchinequality, the intersection of the corresponding hyperplane with P is empty (i.e.,the inequality is strict at any x ∈ P ). We set

Fi =x ∈ P : 〈a i, x 〉+ bi = 0

.

If presentation (1.1) is generic, then each Fi is either a facet of P or is empty.The polar set of a polyhedron P ⊂ Rn is defined as

(1.4) P ∗ =u ∈ Rn : 〈u , x 〉+ 1 > 0 for all x ∈ P

.

The set P ∗ is a convex polyhedron with 0 ∈ P ∗.

Remark. P ∗ is naturally a subset in the dual space (Rn)∗, but we shall notmake this distinction until later on, assuming Rn to be Euclidean. Also, in convexgeometry the inequality 〈u , x 〉 6 1 is usually used in the definition of polarity, butthe definition above is better suited for applications in toric geometry. These twoways of defining the polar set are taken into each other by the central symmetry.

The following properties are well known in convex geometry:

Theorem 1.1.5 (see [42, §2.9] or [325, Th. 2.11]).

(a) P ∗ is bounded if and only if 0 ∈ intP ;(b) P ⊂ (P ∗)∗, and (P ∗)∗ = P if and only if 0 ∈ P ;(c) if a polytope Q is given as a convex hull, Q = conv(a1, . . . ,am), then Q∗

is given by inequalities (1.1) with bi = 1 for 1 6 i 6 m; in particular, Q∗

is a convex polyhedron, but not necessarily bounded;(d) if a polytope P is given by inequalities (1.1) with bi = 1, then P ∗ =

conv(a1, . . . ,am); furthermore, 〈ai, x 〉 + 1 > 0 is a redundant inequalityif and only if ai ∈ conv(aj : j 6= i).

Remark. A polyhedron P admits a presentation (1.1) with bi = 1 if and onlyif 0 ∈ intP . In general, (P ∗)∗ = conv(P,0).


Example 1.1.6. The difference between the situations 0 ∈ P and 0 ∈ intP maybe illustrated by the following example. Let Q = conv(0, e1, e2) be the standard2-simplex in R2. By Theorem 1.1.5 (c), Q∗ is specified by the three inequalities

〈0, x 〉+ 1 > 0, 〈e1, x 〉+ 1 > 0, 〈e2, x 〉+ 1 > 0,

of which the first is satisfied for all x , so we obtain an unbounded polyhedron. Itsdual is conv(0, e1, e2) by Theorem 1.1.5 (d), giving back the standard 2-simplex.

Any combinatorial polytope P has a presentation (1.1) with bi = 1 (take theorigin to the interior of P by a parallel transform, an then divide each of theinequalities in (1.1) by the corresponding bi). Then P ∗ is also a polytope with0 ∈ P ∗, and (P ∗)∗ = P . We refer to the combinatorial polytope P ∗ as the dual ofthe combinatorial polytope P . (We shall not introduce a new notation for the dualpolytope, keeping in mind that polarity is a convex-geometric notion, while dualityof polytopes is combinatorial.)

Theorem 1.1.7 (see [42, §2.10]). If P and P ∗ are dual polytopes, then the faceposet of P ∗ is obtained from the face poset of P by reversing the inclusion relation.

As a corollary, we obtain that P is simple if and only if its dual polytope P ∗ issimplicial. Any polygon is both simple and simplicial.

Proposition 1.1.8. In dimensions n > 3 a simplex is the only polytope whichis both simple and simplicial.

Proof. Let be P be a polytope which is both simple and simplicial. Choose avertex v ∈ P . Since P is simple, v is connected by edges to exactly n other vertices,say v1, . . . , vn. We claim that there are no other vertices in P . To prove this itis enough to show that all vertices v1, . . . , vn are pairwise connected by edges, asthen every vertex from v, v1, . . . , vn will be connected to the remaining n vertices.Indeed, take a pair vi, vj . Since P is simple and v is connected to both vi and vj ,all these three vertices belong to a 2-face. Since P is simplicial and n > 3, this faceis a 2-simplex, in which vi and vj are connected by an edge. We conclude that Phas n+ 1 vertices, so it is an n-simplex.

The proof above also shows that if all 2-faces of a simple polytope are triangular,then P is a simplex. A similar property is valid for a cube: if all 2-faces of a simplepolytope P are quadrangular, then P is a cube (an exercise).

Example 1.1.9. The dual of a simplex is again a simplex. To describe thedual of a cube, we consider the cube [−1, 1]n (the standard cube (1.3) is not goodas 0 is not in its interior). Then the polar set is the convex hull of the endpointsof 2n vectors ±ek, 1 6 k 6 n. It is called the cross-polytope. The 3-dimensionalcross-polytope is the octahedron.

A combinatorial polytope P is called self-dual if P ∗ is combinatorially equiva-lent to P . There are many examples of self-dual non-simple polytopes; an infinitefamily of them is given by k-gonal pyramids for k > 4. Here is a more interestingregular example:

Example 1.1.10 (24-cell). Let Q be the 4-polytope obtained by taking theconvex hull of the following 24 points in R4: endpoints of 8 vectors ±ei, 1 6 i 6 4,


and 16 points of the form (± 12 ,± 1

2 ,± 12 ,± 1

2 ). By Theorem 1.1.5 (c), the polarpolytope Q∗ is given by the following 24 inequalities:

(1.5) ± xi + 1 > 0 for i = 1, . . . , 4, and 12 (± x1 ± x2 ± x3 ± x4) + 1 > 0.

Each of these inequalities turns into equality in exactly one of the specified 24points, so it defines a supporting hyperplane whose intersection with Q is only onepoint. This implies that Q has exactly 24 vertices. The vertices of Q∗ may bedetermined by applying the ‘elimination process’ to (1.5) (see [325, §1.2]), and asa result we obtain 24 points of the form ± e i ± ej for 1 6 i < j 6 4. Eachsupporting hyperplane defined by (1.5) contains exactly 6 vertices of Q∗, whichform an octahedron. So both Q and Q∗ have 24-vertices and 24 octahedral facets.In fact, both Q and Q∗ provide examples of a regular 4-polytope called a 24-cell. Itis the only regular self-dual polytope different from a simplex. For more details onthe 24-cell and other regular polytopes see [85].

Products, hyperplane cuts and connected sums.

Construction 1.1.11 (Product). The product P1×P2 of two simple polytopesP1 and P2 is again a simple polytope. The dual operation on simplicial polytopescan be described as follows. Let S1 ⊂ Rn1 and S2 ⊂ Rn2 be two simplicial polytopes.Assume that both S1 and S2 contain 0 in their interiors. Now define

S1 S2 = conv(S1 × 0 ∪ 0× S2

)⊂ Rn1+n2 .

Then S1 S2 is again a simplicial polytope. For any two simple polytopes P1, P2

containing 0 in their interiors the following identity holds:

P ∗1 P ∗

2 = (P1 × P2)∗.

Both operations × and are also defined on combinatorial polytopes; in this casethe above formula holds without any restrictions.

Construction 1.1.12 (Hyperplane cuts and face truncations). Assume givena simple polytope (1.1) and a hyperplane H = x ∈ Rn : 〈a , x 〉+ b = 0 that doesnot contain any vertex of P . Then the intersections P ∩H> and P ∩H6 of P witheither of the halfspaces determined by H are simple polytopes; we refer to them ashyperplane cuts of P . To see that P ∩ H> and P ∩ H6 are simple we note thattheir new vertices are transverse intersections of H with the edges of P . Since Pis simple, each of those edges is contained in n − 1 facets, so each new vertex ofP ∩H> or P ∩H6 is contained in exactly n facets.

If H separates all vertices of a certain i-face G ⊂ P from the other vertices of Pand G ⊂ H>, then P ∩H> is combinatorially equivalent to G×∆n−i (an exercise),and we say that the polytope P ∩H6 is obtained from P by a face truncation.

In particular, if the cut off face G is a vertex, the result is a vertex truncationof P . When the choice of the cut off vertex is clear or irrelevant we use the notationvt(P ). We also use the notation vtk(P ) for a polytope obtained from P by a k-folditeration of the vertex truncation.

We can describe the face poset of the simple polytope P obtained by truncatingP at a face G ⊂ P as follows. Let F1, . . . , Fm be the facets of P , and assume that

G = Fi1 ∩ · · · ∩ Fik . The polytope P has m facets corresponding to F1, . . . , Fm(and obtained from them by truncation), which we denote by the same letters for


simplicity, and a new facet F = P ∩H . Then we have

Fj1 ∩ · · · ∩ Fjℓ 6= ∅ in P

⇐⇒ Fj1 ∩ · · · ∩ Fjℓ 6= ∅ in P, and Fj1 ∩ · · · ∩ Fjℓ 6⊂ G,F ∩ Fj1 ∩ · · · ∩ Fjℓ 6= ∅ in P

⇐⇒ G ∩ Fj1 ∩ · · · ∩ Fjℓ 6= ∅ in P, and Fj1 ∩ · · · ∩ Fjℓ 6⊂ G.Note that Fj1 ∩ · · · ∩ Fjℓ 6⊂ G if and only if i1, . . . , ik 6⊂ j1, . . . , jℓ.

The two previous construction worked for geometric polytopes. Here is anexample of a construction which is more suitable for combinatorial ones.

Construction 1.1.13 (Connected sum of polytopes). Suppose we are giventwo simple polytopes P and Q, both of dimension n, with distinguished verticesv and w respectively. An informal way to obtain the connected sum P #v,w Q ofP at v and Q at w is as follows. We cut off v from P and w from Q; then, aftera projective transformation, we can glue the rest of P to the rest of Q along thenew simplex facets to obtain P #v,w Q. A more formal definition is given below,following [62, §6].

First, we introduce an n-dimensional polyhedron Γ , which will be used as atemplate for the construction; it arises by considering the standard (n− 1)-simplex∆n−1 in the subspace x : x1 = 0 ⊂ Rn, and taking its cartesian product with thefirst coordinate axis. The facets Gr of Γ therefore have the form R × Dr, whereDr, 1 ≤ r ≤ n, are the facets of ∆n−1. Both Γ and the Gr are divided into positiveand negative halves, determined by the sign of the coordinate x1.

We order the facets of P meeting in v as E1, . . . , En, and the facets of Qmeetingin w as F1, . . . , Fn. Denote the complementary sets of facets by Cv and Cw; thosein Cv avoid v, and those in Cw avoid w.

We now choose projective transformations ϕP and ϕQ of Rn, whose purpose isto map v and w to the infinity of the x1 axis. We insist that ϕP embeds P in Γ so asto satisfy two conditions; firstly, that the hyperplane defining Er is identified withthe hyperplane defining Gr, for each 1 ≤ r ≤ n, and secondly, that the images ofthe hyperplanes defining Cv meet Γ in its negative half. Similarly, ϕQ identifies thehyperplane defining Fr with that defining Gr, for each 1 ≤ r ≤ n, but the imagesof the hyperplanes defining Cw meet Γ in its positive half. We define the connectedsum P #v,w Q of P at v and Q at w to be the simple n-polytope determined bythe images of the hyperplanes defining Cv and Cw and hyperplanes defining Gr,1 6 r 6 n. It is defined only up to combinatorial equivalence; moreover, differentchoices for either of v and w, or either of the orderings for Er and Fr, are likelyto affect the combinatorial type. When the choices are clear, or their effect on theresult irrelevant, we use the abbreviation P #Q.

The related construction of connected sum P # S of a simple polytope P anda simplicial polytope S is described in [325, Example 8.41].

Example 1.1.14.1. If P is an m1-gon and Q is an m2-gon then P #Q is an (m1 +m2 − 2)-gon.2. If P is an n-simplex, then P #Q ≈ vt(P ).3. If both P and Q are n-simplices, then P #Q ≈ vt(∆n) ≈ ∆n−1 ×∆1. The

combinatorial type of vt(∆n) does not depend on the choice of the cut off vertex.All the vertices of the resulting polytope ∆n−1 ×∆1 are equivalent, therefore, the


combinatorial type of vt2(∆n) ≈ ∆n#∆n#∆n is still independent of the choices.The choice of the cut off vertex becomes significant from the next step, i.e. forvt3(∆n), see Exercise 1.1.25.

Neighbourly polytopes.

Definition 1.1.15. A polytope is called k-neighbourly if any set of its k or fewervertices spans a face. According to Exercise 1.1.26, the only n-polytope which ismore than

[n2

]-neighbourly is a simplex. An n-polytope which is

[n2

]-neighbourly is

called simply neighbourly.

Example 1.1.16 (neighbourly 4-polytope). Let P = ∆2 ×∆2, the product oftwo triangles. Then P is simple, and it is easy to see that any two facets of P sharea common 2-face. Therefore, any two vertices of P ∗ are connected by an edge, soP ∗ is a neighbourly simplicial 4-polytope.

More generally, if P ∗1 is k1-neighbourly and P ∗

2 is k2-neighbourly, then (P1 ×P2)

∗ is min(k1, k2)-neighbourly. It follows that (∆n ×∆n)∗ and (∆n ×∆n+1)∗ areneighbourly 2n- and (2n + 1)-polytopes respectively. The next example gives aneighbourly polytope with an arbitrary number of vertices.

Example 1.1.17 (cyclic polytopes). The moment curve in Rn is given by

x : R −→ Rn, t 7→ x (t) = (t, t2, . . . , tn) ∈ Rn.

For any m > n define the cyclic polytope Cn(t1, . . . , tm) as the convex hull of mdistinct points x (ti), t1 < t2 < · · · < tm, on the moment curve.

Theorem 1.1.18.

(a) Cn(t1, . . . , tm) is a simplicial n-polytope;(b) Cn(t1, . . . , tm) has exactly m vertices x(ti), 1 6 i 6 m;(c) the combinatorial type of Cn(t1, . . . , tm) does not depend on the specific

choice of parameters t1, . . . , tm;(d) Cn(t1, . . . , tm) is a neighbourly polytope.

Proof. This proof is taken from [325, Theorem 0.7]. Recall the well-knownVandermonde determinant identity

det

(1 1 · · · 1

x (t0) x (t1) · · · x (tn)

)=

∣∣∣∣∣∣∣∣∣∣∣

1 1 · · · 1t0 t1 · · · tn...

.... . .

...tn−10 tn−1

1 · · · tn−1n

tn0 tn1 · · · tnn

∣∣∣∣∣∣∣∣∣∣∣

=∏

06i<j6n

(tj − ti).

This implies that no n+ 1 points on the moment curve belong to a common affinehyperplane, proving (a). Denote [m] = 1, . . . ,m. Properties (b) and (c) followfrom the following statement: an n-element subset ω ⊂ [m] corresponds to thevertex set of a facet of Cn(t1, . . . , tm) if and only if the following ‘Gale’s evennesscondition’ is satisfied:

If elements i < j are not in ω, then the number of elements k ∈ ωbetween i and j is even.

To prove this we write ω = i1, . . . , in and consider the hyperplane Hω throughthe corresponding points x (tis), 1 6 s 6 n, on the moment curve. We have

Hω = x ∈ Rn : Fω(x ) = 0,


where

Fω(x ) = det

(1 1 . . . 1x x (ti1) . . . x (tin)

).

(The latter is exactly the linear function vanishing on the prescribed points.) Nowlet the point x (t) move on the moment curve. Then Fω(x (t)) is a polynomial in tof degree n. It has n different roots ti1 , . . . , tin , and changes sign at each of them.Now ω corresponds to the vertex set of a facet if and only if Fω(x (ti)) has the samesign for all the points x (ti) with i /∈ ω; that is, if Fω(x (t)) has an even number ofsign changes between t = ti and t = tj , for i < j and i, j /∈ ω. This proves Gale’scondition, and statements (b) and (c).

It remains to prove (d). We need to check that any subset τ = i1, . . . , ik ⊂ [m]of cardinality k 6

[n2

]corresponds to the vertex set of a face. Choose some ε > 0

so that ti < ti + ε < ti+1 for all i < m, and some N > tm + ε. Define a linearfunction Fτ (x ) as

det(x , x (ti1), x (ti1 + ε), . . . , x (tik), x (tik + ε), x (N + 1), . . . , x (N + n− 2k)

).

It vanishes on x (ti) with i ∈ τ . Now Fτ (x (t)) is a polynomial in t of degree n, andit has n different roots

ti1 , ti1 + ε, . . . , tik , tik + ε,N + 1, . . . , N + n− 2k.

If i, j /∈ τ , then there is an even number of roots between t = ti and t = tj , becausea root t = tl always come in a pair with the root t = tl+ε. Thus, the linear functionFτ (x ) has the same sign on all the points x (ti) with i /∈ τ . This linear functiondefines a supporting hyperplane, so τ corresponds to the vertex set of a face.

We shall denote the combinatorial cyclic n-polytope with m vertices by Cn(m).

The vertices and edges of a polytope P determine a graph, which is called thegraph of polytope and denoted Γ(P ). This graph is simple, that is, it has no loopsand multiple edges. The following theorem is due to Blind and Mani, see also [325,§3.4] for a simpler proof given by Kalai.

Theorem 1.1.19. The combinatorial type of a simple polytope P is determinedby its graph Γ(P ). In other words, two simple polytopes are combinatorially equiv-alent if their graphs are isomorphic.

This theorem fails for general polytopes: the graph of a neighbourly polytopeis isomorphic to that of a simplex with the same number of vertices. In general,simplicial n-polytopes are determined by their [n2 ]-skeleta. General n-polytopes aredetermined by their (n−2)-skeleta. See [325, §3.4] for more history and references.

Exercises.

1.1.20. Show that if P and Q are n-polytopes, and the face poset of P is asubposet of the face poset of Q, then P and Q are combinatorially equivalent.

1.1.21. The polyhedron P defined by (1.1) has a vertex if and only if the vectorsa1, . . . ,am span the whole Rn.

1.1.22. Show that a simple n-polytope all of whose 2-faces are quadrangular iscombinatorially equivalent to an n-cube.

1.2. GALE DUALITY AND GALE DIAGRAMS 15

1.1.23. Show that any hyperplane cut of ∆n is combinatorially equivalent toa product of two simplices. Conclude that any combinatorial simple n-polytopewith n+ 2 facets is combinatorially equivalent (in fact, projectively equivalent) toa product of two simplices.

1.1.24. Let P be a simple polytope. Show that if a hyperplane H separates allvertices of a certain i-face G ⊂ P from the other vertices of P and G ⊂ H>, thenP ∩H> ≈ G×∆n−i.

1.1.25. How many combinatorially different polytopes may be obtainedas vt3(∆n)?

1.1.26. Show that if a polytope is k-neighbourly, then every (2k − 1)-face isa simplex. Conclude that if an n-polytope is (

[n2

]+ 1)-neighbourly, then it is a

simplex. Conclude also that a neighbourly 2k-polytope is simplicial. Are thereneighbourly non-simplicial polytopes of odd dimension?

1.1.27. Are the polytopes (∆n×∆n)∗ and (∆n×∆n+1)∗ combinatorially equiv-alent to cyclic polytopes?

1.2. Gale duality and Gale diagrams

The following construction realises any polytope (1.1) of dimension n by theintersection of the orthant

(1.6) Rm> =(y1, . . . , ym) ∈ Rm : yi > 0 for i = 1, . . . ,m

⊂ Rm

with an affine n-plane.

Construction 1.2.1. Let (1.1) be a presentation of a polyhedron. Considerthe linear map A : Rm → Rn sending the ith basis vector e i to a i. It is givenby the n ×m-matrix (which we also denote by A) whose columns are the vectorsai written in the standard basis of Rn. The dual map A∗ : Rn → Rm is given byx 7→ (〈a1, x〉, . . . , 〈am, x 〉). Note that A is of rank n if and only if the polyhedronP has a vertex (e.g., when P is a polytope, see Exercise 1.1.21). Also, let b =(b1, . . . , bm)t ∈ Rm be the column vector of bis. Then we can write (1.1) as

P = P (A, b) =x ∈ Rn : (A∗x + b)i > 0 for i = 1, . . . ,m,

where x = (x1, . . . , xn)t is the column of coordinates. Consider the affine map

iA,b : Rn → Rm, iA,b(x ) = A∗x + b =

(〈a1, x 〉+ b1, . . . , 〈am, x 〉+ bm

)t.

If P has a vertex, then the image of Rn under iA,b is an n-dimensional affine planein Rm, which we can write by m− n linear equations:

(1.7)iA,b(R

n) = y ∈ Rm : y = A∗x + b for some x ∈ Rn= y ∈ Rm : Γy = Γb,

where Γ = (γjk) is an (m−n)×m-matrix whose rows form a basis of linear relationsbetween the vectors a i. That is, Γ is of full rank and satisfies the identity ΓA∗ = 0.

The polytopes P and iA,b(P ) = Rm> ∩ iA,b(Rn) are affinely equivalent.

Example 1.2.2. Consider the standard n-simplex ∆n ⊂ Rn, see (1.2). It isgiven by (1.1) with a i = ei (the ith standard basis vector) for i = 1, . . . , n and


an+1 = −e1−· · ·−en; b1 = · · · = bn = 0 and bn+1 = 1. We may take Γ = (1, . . . , 1)in Construction 1.2.1. Then Γy = y1 + · · ·+ ym, Γb = 1, and we have

iA,b(∆n) =

y ∈ Rn+1 : y1 + · · ·+ yn+1 = 1, yi > 0 for i = 1, . . . , n

.

This is the regular n-simplex in Rn+1.

Construction 1.2.3 (Gale duality). Let a1, . . . ,am be a configuration of vec-tors that span the whole Rn. Form an (m− n) ×m-matrix Γ = (γjk) whose rowsform a basis in the space of linear relations between the vectors a i. The set ofcolumns γ1, . . . , γm of Γ is called a Gale dual configuration of a1, . . . ,am. Thetransition from the configuration of vectors a1, . . . ,am in Rn to the configurationof vectors γ1, . . . , γm in Rm−n is called the (linear) Gale transform. Each configu-ration determines the other uniquely up to isomorphism of its ambient space. Inother words, each of the matrices A and Γ determines the other uniquely up tomultiplication by an invertible matrix from the left.

Using the coordinate-free notation, we may think of a1, . . . ,am as a set oflinear functions on an n-dimensional space W . Then we have an exact sequence

0→WA∗

−→ RmΓ−→ L→ 0,

where A∗ is given by x 7→(〈a1, x 〉, . . . , 〈am, x 〉

), and the map Γ takes ei to

γi ∈ L ∼= Rm−n. Similarly, in the dual exact sequence

0→ L∗ Γ∗

−→ RmA−→W ∗ → 0,

the map A takes e i to a i ∈ W ∗ ∼= Rn. Therefore, two configurations a1, . . . ,amand γ1, . . . , γm are Gale dual if they are obtained as the images of the standardbasis of Rm under the maps A and Γ in a pair of dual short exact sequences.

Here is an important property of Gale dual configurations:

Theorem 1.2.4. Let a1, . . . ,am and γ1, . . . , γm be Gale dual configurations ofvectors in Rn and Rm−n respectively, and let I = i1, . . . , ik ⊂ [m]. Then thesubset ai : i ∈ I is linearly independent if and only if the subset γj : j /∈ I spansthe whole Rm−n.

The proof uses an algebraic lemma:

Lemma 1.2.5. Let k be a field or Z, and assume given a diagram

0↓Uyi1

0 −−−−→ Ri2−−−−→ S

p2−−−−→ T −−−−→ 0yp1V↓0

in which both vertical and horizontal lines are short exact sequences of k-vectorspaces or free abelian groups. Then p1i2 is surjective (respectively, injective or splitinjective) if and only if p2i1 is surjective (respectively, injective or split injective).


Proof. This is a simple diagram chase. Assume first that p1i2 is surjective.Take α ∈ T ; we need to cover it by an element in U . There is β ∈ S such thatp2(β) = α. If β ∈ i1(U), then we are done. Otherwise set γ = p1(β) 6= 0. Sincep1i2 is surjective, we can choose δ ∈ R such that p1i2(δ) = γ. Set η = i2(δ) 6= 0.Hence, p1(η) = p1(β)(= γ) and there is ξ ∈ U such that i1(ξ) = β − η. Thenp2i1(ξ) = p2(β − η) = α− p2i2(δ) = α. Thus, p2i1 is surjective.

Now assume that p1i2 is injective. Suppose p2i1(α) = 0 for a nonzero α ∈ U .Set β = i1(α) 6= 0. Since p2(β) = 0, there is a nonzero γ ∈ R such that i2(γ) = β.Then p1i2(γ) = p1(β) = p1i1(α) = 0. This contradicts the assumption that p1i2 isinjective. Thus, p2i1 is injective.

Finally, if p1i2 is split injective, then its dual map i∗2p∗1 : V

∗ → R∗ is surjective.Then i∗1p

∗2 : T

∗ → U∗ is also surjective. Thus, p2i1 is split injective.

Proof of Theorem 1.2.4. Let A be the n×m-matrix with column vectorsa1, . . . ,am, and let Γ be the (m−n)×m-matrix with columns γ1, . . . , γm. Denoteby AI the n×k-submatrix formed by the columns a i : i ∈ I and denote by ΓI the(m−n)× (m− k)-submatrix formed by the columns γj : j /∈ I. We also considerthe corresponding maps AI : Rk → Rn and ΓI : R

m−k → Rm−n.

Let i : Rk → Rm be the inclusion of the coordinate subspace spanned by thevectors ei, i ∈ I, and let p : Rm → Rm−k the projection sending every such eito zero. Then AI = A · i and Γ ∗

I= p · Γ ∗. The vectors a i : i ∈ I are linearly

independent if and only if AI = A · i is injective, and the vectors γj : j /∈ Ispan Rm−n if and only if Γ ∗

I= p·Γ ∗ is injective. These two conditions are equivalent

by Lemma 1.2.5.

Construction 1.2.6 (Gale diagram). Let P be a polytope (1.1) with bi = 1

and let P ∗ = conv(a1, . . . ,am) be the polar polytope. Let A∗ = (A∗1) be the

m × (n + 1)-matrix obtained by appending a column of units to A∗. The matrix

A∗ has full rank n+ 1 (indeed, otherwise there is x ∈ Rn such that 〈a i, x〉 = 1 forall i, and then λx is in P for any λ > 1, so that P is unbounded). A configuration

of vectors G = (g1, . . . , gm) in Rm−n−1 which is Gale dual to A is called a Galediagram of P ∗. A Gale diagram G = (g1, . . . , gm) of P ∗ is therefore determined bythe conditions

GA∗ = 0, rankG = m− n− 1, and

m∑

i=1

g i = 0.

The rows of the matrix G from a basis of affine dependencies between the vectorsa1, . . . ,am, i.e. a basis in the space of y = (y1, . . . , ym)

t satisfying

y1a1 + · · ·+ ymam = 0, y1 + · · ·+ ym = 0.

Proposition 1.2.7. The polyhedron P = P (A, b) is bounded if and only if thematrix Γ = (γjk) can be chosen so that the affine plane iA,b(Rn) is given by

(1.8) iA,b(Rn) =

y ∈ Rm : γ11y1 + · · ·+ γ1mym = c,

γj1y1 + · · ·+ γjmym = 0, 2 6 j 6 m− n.

,

where c > 0 and γ1k > 0 for all k.Furthermore, if bi = 1 in (1.1), then the vectors gi = (γ2i, . . . , γm−n,i)

t, i =1, . . . ,m, form a Gale diagram of the polar polytope P ∗ = conv(a1, . . . ,am).


Proof. The image iA,b(P ) is the intersection of the n-plane L = iA,b(Rn) withRm> . It is bounded if and only if L0∩Rm> = 0, where L0 is the n-plane through 0

parallel to L. Choose a hyperplaneH0 through 0 such that L0 ⊂ H0 and H0∩Rm> =

0. Let H be the affine hyperplane parallel to H0 and containing L. Since L ⊂ H ,we may take the equation defining H as the first equation in the system Γy = Γb

defining L. The conditions on H0 imply that H ∩ Rm> is nonempty and bounded,that is, c > 0 and γ1k > 0 for all k. Now, subtracting the first equation from theother equations of the system Γy = Γb with appropriate coefficients, we achievethat the right hand sides of the last m− n− 1 equations become zero.

To prove the last statement, we observe that in our case

Γ =

(γ11 · · · γ1mg1 · · · gm

).

The conditions ΓAt = 0 and rankΓ = m − n imply that GAt = 0 and rankG =

m − n − 1. Finally, by comparing (1.7) with (1.8) we obtain Γb =

(c0

). Since

bi = 1, we get∑m

i=1 g i = 0. Thus, G = (g1, . . . , gm) is a Gale diagram of P ∗.

Corollary 1.2.8. A polyhedron P = P (A, b) is bounded if and only if thevectors a1, . . . ,am satisfy α1a1 + · · ·+ αmam = 0 for some positive numbers αk.

Proof. If a1, . . . ,am satisfy∑mk=1 αkak = 0 with positive αk, then we can

take∑mk=1 αkyk =

∑mk=1 αkbk as the first equation defining the n-plane iA,b(Rn)

in Rm. It follows that iA,b(P ) is contained in the intersection of the hyperplane∑mk=1 αkyk =

∑mk=1 αkbk with Rm> , which is bounded since all αk are positive.

Therefore, P is bounded.Conversely, if P is bounded, then it follows from Proposition 1.2.7 and Gale

duality that a1, . . . ,am satisfy γ11a1 + · · ·+ γ1mam = 0 with γ1k > 0.

A Gale diagram of P ∗ encodes its combinatorics (and the combinatorics of P )completely. We give the corresponding statement in the generic case only:

Proposition 1.2.9. Assume that (1.1) is a generic presentation with bi = 1.Let P ∗ = conv(a1, . . . ,am) be the polar simplicial polytope and G = (g1, . . . , gm)be its Gale diagram. Then the following conditions are equivalent:

(a) Fi1 ∩ · · · ∩ Fik 6= ∅ in P = P (A,1);(b) conv(ai1 , . . . ,aik) is a face of P ∗;(c) 0 ∈ conv

(gj : j /∈ i1, . . . , ik

).

Proof. The equivalence (a)⇔ (b) follows from Theorems 1.1.5 and 1.1.7.(b)⇒ (c). Let conv(a i1 , . . . ,a ik) be a face of P ∗. We first observe that each of

ai1 , . . . ,a ik is a vertex of this face, as otherwise presentation (1.1) is not generic.By definition of a face, there exists a linear function ξ such that ξ(aj) = 0 forj ∈ i1, . . . , ik and ξ(aj) > 0 otherwise. The condition 0 ∈ intP ∗ implies thatξ(0) > 0, and we may assume that ξ has the form ξ(u) = 〈u , x 〉 + 1 for somex ∈ Rn. Set yj = ξ(a j) = 〈aj , x 〉+ 1, i.e. y = A∗x + 1. We have

∑

j /∈i1,...,ik

g jyj =

m∑

j=1

g jyj = Gy = G(A∗x + 1) = G1 =

m∑

j=1

g j = 0,

where yj > 0 for j /∈ i1, . . . , ik. It follows that 0 ∈ conv(g j : j /∈ i1, . . . , ik).


(c)⇒ (b). Let∑j /∈i1,...,ik

g jyj = 0 with yj > 0 and at least one yj nonzero.

This is a linear relation between the vectors g j . The space of such linear relations

has basis formed by the columns of the matrix A∗ = (A∗1) Hence, there exists

x ∈ Rn and b ∈ R such that yj = 〈a j , x 〉+ b. The linear function ξ(u) = 〈u , x 〉+ btakes zero values on aj with j ∈ i1, . . . , ik and takes nonnegative values on theother aj . Hence, a i1 , . . . ,a ik is a subset of the vertex set of a face. Since P ∗ issimplicial, a i1 , . . . ,aik is a vertex set of a face.

Remark. We allow redundant inequalities in Proposition (1.2.9). In this casewe obtain the equivalences

Fi = ∅ ⇔ a i ∈ conv(a j : j 6= i) ⇔ 0 /∈ conv(g j : j 6= i).

A configuration of vectors G = (g1, . . . , gm) in Rm−n−1 with the property

0 ∈ conv(gj : j /∈ i1, . . . , ik

)⇔ conv(a i1 , . . . ,a ik) is a face of P ∗

is called a combinatorial Gale diagram of P ∗ = conv(a1, . . . ,am). For example, aconfiguration obtained by multiplying each vector in a Gale diagram by a positivenumber is a combinatorial Gale diagram. Also, the vectors of a combinatorial Galediagram can be moved as long as the origin does not cross the ‘walls’, i.e. affinehyperplanes spanned by subsets of g1, . . . , gm. A combinatorial Gale diagram ofP ∗ is a Gale diagram of a polytope which is combinatorially equivalent to P ∗.

Example 1.2.10.1. The Gale diagram of ∆n (when m = n+1) consists of n+1 points 0 in R0,

i.e. g i = 0, i = 1, . . . ,m.2. Let P = ∆p−1 ×∆q−1, p+ q = m, i.e. m = n+ 2. Then P ∗ is a hyperplane

cut of a simplex ∆m−2 by a hyperplane that separates some p−1 of its vertices fromthe other q − 1. A combinatorial Gale diagram of P ∗ consists of p points 1 ∈ R1

and q points −1 ∈ R1. The cases p = 1 or q = 1 correspond to a presentation of∆m−2 with one redundant inequality.

3. A combinatorial Gale diagram of a pentagon (m = n+3) is shown in Fig. 1.1.The property that conv(a1,a2) is a face translates to 0 ∈ conv(g3, g4, g5).

a5

a1

a2

a3

a4

rrrrr

❩

❩❩❩❩

g3

g1

g4

g2

g5

rrrrr

❩

❩❩❩❩

r0

Figure 1.1. A pentagon and its Gale diagram.

Gale diagrams provide an efficient tool for studying the combinatorics of higher-dimensional polytopes with few vertices, as in this case a Gale diagram translatesthe higher-dimensional structure to a low-dimensional one. For example, Galediagrams are used to classify n-polytopes with up to n + 3 vertices and to findunusual examples when the number of vertices is n+ 4, see [325, §6.5].


Exercises.

1.2.11. Describe combinatorial Gale diagrams of polytopes shown in Fig. 1.2.

1.3. Face vectors and Dehn–Sommerville relations

The notion of the f -vector (or face vector) is a central concept in the combi-natorial theory of polytopes. It has been studied since the time of Euler.

Definition 1.3.1. Let P be a convex n-polytope. Denote by fi the numberof i-dimensional faces of P . The integer sequence f (P ) = (f0, f1, . . . , fn) is knownas the f -vector (or the face vector) of P . Note that fn = 1. The homogeneousF -polynomial of P is defined by

F (P )(s, t) = sn + fn−1sn−1t+ · · ·+ f1st

n−1 + f0tn.

The h-vector h(P ) = (h0, h1, . . . , hn) and the H-polynomial of P are defined by

(1.9)h0s

n + h1sn−1t+ · · ·+ hnt

n = (s− t)n + fn−1(s− t)n−1t+ · · ·+ f0tn,

H(P )(s, t) = h0sn + h1s

n−1t+ · · ·+ hn−1stn−1 + hnt

n = F (P )(s− t, t).The g-vector of a simple polytope P is the vector g(P ) = (g0, g1, . . . , g[n

2

]), where

g0 = 1 and gi = hi − hi−1 for i = 1, . . . , [n/2].

Example 1.3.2. We have

F (∆n) = sn +

(n+ 1

1

)sn−1t+

(n+ 1

2

)sn−2t2 + · · ·+ tn =

(s+ t)n+1 − tn+1

s,

H(∆n) = sn + sn−1t+ sn−2t2 + · · ·+ tn =sn+1 − tn+1

s− t .

Obviously, the f -vector is a combinatorial invariant of a polytope, that is, itdepends only on the face poset. This invariant is far from being complete, even forsimple polytopes:

Example 1.3.3. Two different combinatorial simple polytopes may have samef -vectors. For instance, let P be the 3-cube and Q a simple 3-polytope with 2triangular, 2 quadrangular and 2 pentagonal facets, see Figure 1.2. (Note that Qis obtained by truncating a tetrahedron at two vertices, it is also dual to the cyclicpolytope C3(6) from Definition 1.1.17.) Then f (P ) = f (Q) = (8, 12, 6).

❩❩

❩❩

❩❩

PPPPP

❩❩

❩❩

❩❩

Figure 1.2. Two combinatorially non-equivalent simple polytopeswith the same f -vectors.

1.3. FACE VECTORS AND DEHN–SOMMERVILLE RELATIONS 21

The f -vector and the h-vector contain equivalent combinatorial information,and determine each other by means of linear relations, namely

(1.10) hk =k∑

i=0

(−1)k−i(n− in− k

)fn−i, fk =

n∑

q=k

(q

k

)hn−q, for 0 6 k 6 n.

In particular, h0 = 1 and hn = f0 − f1 + · · ·+ (−1)nfn. By the Euler formula,

(1.11) f0 − f1 + · · ·+ (−1)nfn = 1,

which is equivalent to hn = h0. This is the first evidence of the fact that manycombinatorial relations for the face numbers have much simpler form when writtenin terms of the h-vector. Another example of this phenomenon is given by thefollowing generalisation of the Euler formula for simple or simplicial polytopes.

Theorem 1.3.4 (Dehn–Sommerville relations). The h-vector of any simple n-polytope is symmetric, that is,

H(s, t) = H(t, s), or hi = hn−i for 0 6 i 6 n.

The Dehn–Sommerville relations can be proved in many different ways. Wepresent a proof from [42], which can be viewed as a combinatorial version of Morse-theoretic arguments. An alternative proof will be given in Section 1.7.

Proof of Theorem 1.3.4. Let P ⊂ Rn be a simple polytope. Choose ageneric linear function ϕ : Rn → R which distinguishes the vertices of P . Writeϕ(x ) = 〈νν, x 〉 for some vector νν in Rn. The assumption on ϕ implies that νν isparallel to no edge of P . We can view ϕ as a height function on P and turn the1-skeleton of P into a directed graph by orienting each edge in such a way that ϕincreases along it, see Figure 1.3. For each vertex v of P define the index indν(v) as

❳❳❳❳❳❳②

❳❳❳❳❳❳②

❳❳❳❳❳❳②

νν

ind = 0

ind = 1

ind = 2

ind = 3

Figure 1.3. Orienting the 1-skeleton of P .

the number of incident edges that point towards v. Denote the number of verticesof index i by Iν(i). We claim that Iν(i) = hn−i. Indeed, each face of P has aunique top vertex (the maximum of the height function ϕ restricted to the face)and a unique bottom vertex (the minimum of ϕ). Let G be a k-face of P , and v

Gits

top vertex. Since P is simple, there are exactly k edges of G meeting at vG, whence

ind(vG) > k. On the other hand, each vertex of index q > k is the top vertex for


exactly(qk

)faces of dimension k. It follows that the number of k-faces of P can be

calculated as

fk =∑

q>k

(q

k

)Iν(q).

Now the second identity from (1.10) shows that Iν(q) = hn−q, as claimed. Inparticular, the number Iν(q) does not depend on νν. At the same time, we haveindν(v) = n− ind−ν(v) for any vertex v, which implies that

hn−q = Iν(q) = I−ν(n− q) = hq.

Remark. The above proof also shows that the numbers hk = Iν(n − k) arenonnegative, which is not evident from (1.10). On the other hand, the nonnegativityof the h-vector translates into certain conditions on the f -vector. This will beimportant in the subsequent study of f -vectors for combinatorial objects moregeneral than simple polytopes.

Theorem 1.3.5. The f -vector of a simple n-polytope satisfies

(1.12) fk =

k∑

i=0

(−1)i(n− in− k

)fi, for 0 6 k 6 n.

Proof. By the Dehn–Sommerville relations,

F (s− t, t) = H(s, t) = H(t, s) = F (t− s, s).By substituting u = s− t we obtain F (u, t) = F (−u, t+ u), or

un + fn−1un−1t+ · · ·+ f1ut

n−1 + f0un

= (−u)n + fn−1(−u)n−1(t+ u) + · · ·+ f1(−u)(t+ u)n−1 + f0(t+ u)n.

Calculating the coefficient of uktn−k in both sides above yields (1.12).

By Theorem 1.1.7, the f -vector of the dual n-polytope P ∗ satisfies

fi(P∗) = fn−1−i(P ), for 0 6 i 6 n− 1.

Then it follows from (1.12) that the f -vector of a simplicial polytope satisfies the

relations fn−1−k =∑k

i=0(−1)i(n−in−k

)fn−1−i or, equivalently,

fq−1 =n∑

j=q

(−1)n−j(j

q

)fj−1, for 1 6 q 6 n+ 1.

Proposition 1.3.6. The F - and H-polynomial are multiplicative, i.e.

(1.13) F (P1 × P2) = F (P1)F (P2), H(P1 × P2) = H(P1)H(P2).

for any convex polytopes P1 and P2.

Proof. Let dimP1 = n1 and dimP2 = n2. Each k-face of P1 × P2 is theproduct of an i-face of P1 and a (k − i)-face of P2 for some i, whence

(1.14) fk(P1 × P2) =

n1∑

i=0

fi(P1)fk−i(P2), for 0 6 k 6 n1 + n2.

This implies the first identity, and the second follows from (1.9).

1.3. FACE VECTORS AND DEHN–SOMMERVILLE RELATIONS 23

Example 1.3.7. We have F (In) = (F (∆1))n and H(In) = (H(∆1))n, i.e.

F (In) = (s+ 2t)n, H(In) = (s+ t)n.

We can also express the f -vector and the h-vector of the connected sum P #Qin terms of those of P and Q (the proof is left as an exercise):

Proposition 1.3.8. Let P and Q be simple n-polytopes. Then

f0(P #Q) = f0(P ) + f0(Q)− 2; h0(P #Q) = hn(P #Q) = 1;

fi(P #Q) = fi(P ) + fi(Q)−(n

i

), for 1 6 i 6 n;

hi(P #Q) = hi(P ) + hi(Q), for 1 6 i 6 n− 1.

Using the Dehn–Sommerville relations we can show that a simplicial polytopecannot be ‘too neighbourly’ (see Definition 1.1.15) if it is not a simplex.

Proposition 1.3.9. Let S be a q-neighbourly simplicial n-polytope, and letS 6= ∆n. Then q 6

[n2

].

Proof. Let S∗ be the dual polytope. By Theorem 1.1.7, fn−i(S∗) = fi−1(S) =(

mi

)for 1 6 i 6 q, where m is the number of vertices of S. From (1.10) we get

(1.15) hk(S∗) =

k∑

i=0

(−1)k−i(n− ik − i

)(m

i

)=

(m− n+ k − 1

k

), for k 6 q,

The second equality is obtained by calculating the coefficient of tk on both sides of

1

(1 + t)n−k+1(1 + t)m = (1 + t)m−n+k−1.

If S 6= ∆n, then m > n + 1, which together with (1.15) gives h0(S∗) < h1(S

∗) <· · · < hq(S

∗). It then follows from the Dehn–Sommerville relations that q 6[n2

].

Since the H-polynomial of a simple n-polytope P satisfies the identityH(P )(s, t) = H(P )(t, s), we can express it in terms of elementary symmetricfunctions as follows:

(1.16) H(P )(s, t) =

[n/2]∑

i=0

γi(s+ t)n−2i(st)i.

The identity h0 = hn = 1 implies that γ0 = 1.

Definition 1.3.10. The integer sequence γ(P ) = (γ0, . . . , γ[n/2]) is called theγ-vector of P . We refer to

γ(P )(τ) = γ0 + γ1τ + · · ·+ γ[n/2]τ[n/2]

as the γ-polynomial of P .

Example 1.3.11. We have γ(I1)(τ) = 1. If P 2m is an m-gon, then γ(P 2

m)(τ) =1 + (m− 4)τ .

The components of the γ-vector can be expressed via the components of anyof the f -, h- or g-vector by means of linear relations, and vice versa. The explicittransition formulae between the g- and γ-vectors are given by the next lemma.


Lemma 1.3.12. Let P be a simple n-polytope. Then

gi = (n− 2i+ 1)

i∑

j=0

1

n− i− j + 1

(n− 2j

i− j

)γj ;

γi = (−1)ii∑

j=0

(−1)j(n− i− ji− j

)gj , 0 6 i 6

[n2

].

Proof. Using the formulae of Examples 1.3.2 and 1.3.7 we calculate

H(P ) =

[n/2]∑

j=0

γj(st)jH(In−2j),

H(Iq) =

q∑

j=0

(q

j

)sjtq−j =

[q/2]∑

k=0

ck,q(st)kH(∆q−2k),

where c0,q = 1 and ck,q =(qk

)−(q

k−1

)= q−2k+1

q−k+1

(qk

)> 0 for k = 1, . . . , [ q2 ]. Hence

H(P )(s, t) =

[n/2]∑

i=0

( i∑

j=0

ci−j,n−2jγj

)(st)iH(∆n−2i).

On the other hand,

H(P )(s, t) =

[n/2]∑

i=0

gi(st)iH(∆n−2i).

Comparing the last two formulae we obtain gi =∑i

j=0 ci−j,n−2jγj , which is equiv-alent to the first required formula. The second is left as an exercise.

Proposition 1.3.13. Let P,Q be two simple n-polytopes, and consider thefollowing four conditions:

(a) fi(P ) > fi(Q) for i = 0, 1, . . . , n;(b) hi(P ) > hi(Q) for i = 0, 1, . . . , n;(c) gi(P ) > gi(Q) for i = 0, 1, . . . ,

[n2

];

(d) γi(P ) > γi(Q) for i = 0, 1, . . . ,[n2

].

Then (d)⇒(c)⇒(b)⇒(a).

Proof. The first of equations (1.10) implies that the components of f (P )are expressed via the components of h(P ) with positive coefficients, which proves

the implication (b)⇒(a). We have hk =∑k

i=0 gi for 0 6 k 6[n2

], which gives

the implication (c)⇒(b). The implication (d)⇒(c) follows similarly from the firstformula of Lemma 1.3.12.

The components of the f -vector of any polytope are nonnegative. The nonneg-ativity of the components of the h-vector of a simple polytope follows from theirgeometric interpretation obtained in the proof of Theorem 1.3.4; the h-vector ofa non-simple polytope may have negative components (e.g. for the octahedron).The nonnegativity of the g-vector of a simple n-polytope (that is, the inequalitiesgi(P ) > gi(∆

n)) is a much more subtle property; it follows from the g-theoremdiscussed in the next section. The γ-vector of a simple polytope may have nega-tive components (e.g. for P = ∆2); its nonnegativity for special classes of simple

1.4. CHARACTERISING THE FACE VECTORS OF POLYTOPES 25

polytopes will be discussed in Section 1.6. The nonnegativity of the γ-vector canbe expressed by the inequalities γi(P ) > γi(I

n), see Exercise 1.3.17.

Exercises.

1.3.14. Show that any 3-polytope has a face with 6 5 vertices.

1.3.15. Prove Proposition 1.3.8.

1.3.16. Prove the second transition formula of Lemma 1.3.12.

1.3.17. Show that the γ-polynomial is multiplicative, that is,

γ(P ×Q)(τ) = γ(P )(τ) · γ(Q)(τ)

In particular, γ(In)(τ) = 1, i.e. γ(In) = (1, 0, . . . , 0).

1.4. Characterising the face vectors of polytopes

The face numbers are the simplest combinatorial invariants of polytopes, andthey arise in many hard problems of combinatorial geometry. One of the mostnatural and basic questions is to describe all possible face numbers, or, more pre-cisely, determine which integer vectors arise as the f -vectors of polytopes. In thegeneral case this questions is probably intractable (see the end of this section), buta particularly nice answer exists in the case of simple (or, equivalently, simplicial)polytopes. Obviously, the Dehn–Sommerville relations provide a necessary condi-tion. As far as only linear equations are concerned, there are no further restrictions:

Proposition 1.4.1 (Klee [186]). The Dehn–Sommerville relations are the mostgeneral linear equations satisfied by the face numbers of all simple polytopes.

Proof. Once again, the use of the h-vector simplifies the proof significantly.Given a simple polytope P , we set

h(P )(t) = H(P )(1, t) = h0(P ) + h1(P )t+ · · ·+ hn(P )tn.

It is enough to prove that the affine hull of the h-vectors (h0, h1, . . . , hn) of simplen-polytopes is an

[n2

]-dimensional plane. This can be done by presenting

[n2

]+ 1

simple polytopes with affinely independent h-vectors. Take Qk = ∆k ×∆n−k for0 6 k 6

[n2

]. Since h(∆k)(t) = 1 + t+ · · ·+ tk, formula (1.13) gives

h(Qk)(t) =1− tk+1

1− t · 1− tn−k+1

1− t .

It follows that the lowest degree term in the polynomial h(Qk+1)(t) − h(Qk)(t) istk+1, for 0 6 k 6

[n2

]− 1. Therefore, the vectors h(Qk) are affinely independent

for these values of k.

Example 1.4.2. Let P be a simple polytope. Since every vertex is containedin exactly n edges and each edge connects two vertices, we have a linear relation

(1.17) 2f1 = nf0.

By Proposition 1.4.1 this must be a consequence of the Dehn–Sommerville relations(in fact, it is equation (1.12) for k = 1.)

Equation (1.17) together with the Euler identity (1.11) shows that the f -vectorof a simple 3-polytope P 3 is completely determined by the number of facets, namely,

f (P 3) = (2f2 − 4, 3f2 − 6, f2, 1).


Similarly, the f -vector of a simplicial 3-polytope S3 is determined by the numberof vertices, namely,

f (S3) = (f0, 3f0 − 6, 2f0 − 4, 1).

Remark. Euler’s formula (1.11) is the only linear relation satisfied by the facevectors of general polytopes. This can be proved similarly to Proposition 1.4.1, byspecifying sufficiently many polytopes with affinely independent face vectors.

Apart from the linear equations, the f -vectors of polytopes satisfy certain in-equalities. Here are some of the simplest of them.

Example 1.4.3. There are the following obvious lower bounds for the numberof vertices and the number of facets of an n-polytope:

f0 > n+ 1, fn−1 > n+ 1.

Since every pair of vertices is joined by at most one edge, and every pair of facetsintersect at most one face of codimension 2, we have the upper bounds

f1 6(f02

), fn−2 6

(fn−1

2

).

If the polytope is simplicial, then there is also the following lower bound for f1:

f1 > nf0 −(n+12

).

It is much more difficult to prove though, even for 4-polytopes. For simplicial3-polytopes the inequality above turns into identity.

Historically, the most important inequality-type results preceding the generalcharacterisation of f -vectors were the Upper Bound Theorem (UBT) and the LowerBound Theorem (LBT). They give respectively an upper and a lower bound for thenumber of faces in a simplicial polytope with the given number of vertices.

Theorem 1.4.4 (UBT for simplicial polytopes). Among all simplicial n-polytopes S with m vertices the cyclic polytope Cn(m) (Example 1.1.17) has themaximal number of i-faces for 1 6 i 6 n. That is, if f0(S) = m, then

fi(S) 6 fi(Cn(m)

)for i = 1, . . . , n.

Equality is achieved for all i if and only if S is a neighbourly polytope.

The UBT was conjectured by Motzkin and proved by McMullen [217] in 1970.Since Cn(m) is neighbourly, we have

fi(Cn(m)

)=(mi+1

)for 0 6 i 6

[n2

]− 1.

Due to the Dehn–Sommerville relations, this determines the full f -vector of Cn(m).The exact values are given by the following lemma.

Lemma 1.4.5. The number of i-faces of the cyclic polytope Cn(m) (or anyneighbourly n-polytope with m vertices) is given by

fi =

[n2

]∑

q=0

(q

n− 1− i

)(m− n+ q − 1

q

)+

[n−12

]∑

p=0

(n− p

n− 1− i

)(m− n+ p− 1

p

)

for 0 6 i 6 n, where we set(pq

)= 0 for p < q or q < 0.


Proof. We set C = Cn(m). Using the second identity from (1.10), the identity[n2

]+1 = n−

[n−12

], the Dehn–Sommerville relations for C∗, and (1.15), we calculate

fi(C) = fn−1−i(C∗) =

n∑

q=0

(q

n−1−i

)hn−q(C

∗)

=

[n/2]∑

q=0

(q

n−1−i

)hq(C

∗) +n∑

q=[n/2]+1

(q

n−1−i

)hn−q(C

∗)

=

[n/2]∑

q=0

(q

n−1−i

)(m−n+q−1

q

)+

[(n−1)/2]∑

p=0

(n−pn−1−i

)(m−n+p−1

p

).

Lemma 1.4.6. Assume that the inequalities

hi(P ) 6(m−n+i−1

i

), i = 0, . . . ,

[n2

]

hold for the h-vector of a simple polytope P with m facets. Then the dual simplicialpolytope P ∗ satisfies the UBT inequalities fi(P

∗) 6 fi(Cn(m)

)for i = 1, . . . , n.

Proof. Since hi((Cn(m))∗

)=(m−n+i−1

i

)for i = 0, . . . ,

[n2

], the statement

follows from Proposition 1.3.13. Alternatively, replace the last ‘=’ in the calculationfrom the proof of Lemma 1.4.5 by ‘6’.

This was one of the key observations in McMullen’s original proof [217] of theUBT for simplicial polytopes (the whole proof can also be found in [42, §18] and[325, §8.4]). R. Stanley gave an algebraic argument establishing the inequalitiesfrom Lemma 1.4.6, and therefore the UBT, in a much more general setting oftriangulated spheres. We shall discuss Stanley’s approach and conclude the proofof the UBT in Section 3.3.

Remark. The UBT holds for all convex polytopes. That is, the cyclic polytopeCn(m) has the maximal number of i-faces among all convex n-polytopes with mvertices. The argument for this builds on the following observation of Klee andMcMullen, which we reproduce from [325, Lemma 8.24].

Lemma 1.4.7. By a small perturbation of vertices of an n-polytope P one canachieve that the resulting polytope P ′ is simplicial, and

fi(P ) 6 fi(P′) for i = 1, . . . , n− 1.

Definition 1.4.8. A simplicial n-polytope S is called stacked if there is asequence S0, S1, . . . , Sk = S of n-polytopes such that S0 is an n-simplex and Si+1

is obtained from Si by adding a pyramid over a facet of Si (the vertex of theadded pyramid is chosen close enough to its base, so that the whole constructionremains convex and simplicial). The polar simple polytopes are those obtainedfrom a simplex by iterating the vertex cut operation of Example 1.1.14.2. Theseare sometimes called truncation polytopes.

The f -vector of a stacked polytope is easy to calculate (see Exercise 1.4.17).

Theorem 1.4.9 (LBT for simplicial polytopes). Among all simplicial n-polytopes S with m vertices a stacked polytope has the minimal number of i-faces


for 2 6 i 6 n− 1. That is, if f0(S) = m, then

fi(S) >(ni

)m−

(n+1i+1

)i for i = 1, . . . , n− 2;

fn−1(S) > (n− 1)m− (n+ 1)(n− 2).

For n 6= 3 equality is achieved for all i if and only if S is a stacked polytope.

Remark. For n = 3 the LBT inequalities f1 > 3m− 6 and f2 > 2m− 4 turninto equalities for all simplicial polytopes.

An inductive argument by McMullen, Perles and Walkup [222] reduced the

LBT to the case i = 1, namely, to the inequality f1 > nm−(n+12

). It was finally

proved by Barnette [19], [20]. Barnette’s proof of the LBT, with some simplifica-tions, can also be found in [42]. The fact that equality is achieved only for stackedpolytopes (if n 6= 3) was proved by Billera and Lee [30].

Remark. Unlike the UBT, little is know about generalisations of the LBTto non-simplicial convex polytopes. Some results in this direction were obtainedin [177] along with generalisations of the LBT to triangulated spheres and mani-folds, which we also discuss later in this book.

An easy calculation shows that the inequalities fn−1 > n+ 1 and and fn−2 >

nfn−1 −(n+12

)for simple polytopes from Example 1.4.3 can be written in terms

of the h-vector as follows: h0 6 h1 6 h2. Having reduced the whole LBT to theinequality h1 6 h2, McMullen and Walkup [222] conjectured that the componentsof the h-vector ‘grow up to the middle’, that is the inequalities

(1.18) h0 6 h1 6 · · · 6 h[n2

]

hold for a simple n-polytope. It has since become known as the Generalised LowerBound Conjecture (GLBC).

McMullen also suggested a generalisation to the UBT, whose formulation re-quires an algebraic digression.

Definition 1.4.10. For any two positive integers a, i there exists a uniquebinomial i-expansion of a of the form

a =(aii

)+(ai−1

i−1

)+ · · ·+

(ajj

),

where ai > ai−1 > · · · > aj > j > 1.The binomial i-expansion of a can be constructed by choosing ai as the unique

number satisfying(aii

)6 a <

(ai+1i

), then choosing ai−1, and so on. One needs

only to check that ai > ai−1, which is straightforward.Now define the ith pseudopower of a as

a〈i〉 =(ai+1i+1

)+(ai−1+1

i

)+ · · ·+

(aj+1j+1

), 0〈i〉 = 0.

Example 1.4.11.1. For a > 0, a〈1〉 =

(a+12

).

2. If i > a then the binomial expansion has the form

a =(ii

)+(i−1i−1

)+ · · ·+

(i−a+1i−a+1

)= 1 + · · ·+ 1,

and therefore a〈i〉 = a.3. Let a = 28, i = 4. Then

28 =(64

)+(53

)+(32

), 28〈4〉 =

(75

)+(64

)+(43

)= 40.


The importance of the binomial expansion and pseudopowers comes from thefollowing fundamental result of combinatorial commutative algebra.

Theorem 1.4.12 (Macaulay, Stanley). The following two conditions are equiv-alent for a sequence of integers (k0, k1, k2, . . .):

(a) k0 = 1 and 0 6 ki+1 6 k〈i〉i for i > 1;

(b) there exists a connected commutative graded algebra A = A0⊕A1⊕A2⊕· · ·over a field k such that A is generated by its degree-one elements anddimkA

i = ki for i > 0.

Macaulay’s original theorem [202] says that (b) above is equivalent to theexistence of a multicomplex whose h-vector is given by (k0, k1, k2, . . .). The originalproof is long and complicated. The reformulation of Macaulay’s condition in termsof pseudopowers, i.e. condition (a), is due to Stanley [293, Theorem 2.2]. Simplerproofs of Theorem 1.4.12 can be found in [79] and [45, §4.2].

Definition 1.4.13. A sequence of integers satisfying either of the conditionsof Theorem 1.4.12 is called an M-sequence. Finite M -sequences are M-vectors.

Now observe that the number(m−n+i−1

i

)on the right hand side of the inequality

of Lemma 1.4.6 equals the number of degree i monomials in h1 = m−n generators.It follows that the inequalities of Lemma 1.4.6, and therefore the UBT, hold if theh-vector is an M -vector.

In 1970 McMullen [218] combined all known and conjectured information aboutthe f -vectors, including the Dehn–Sommerville relations and the generalisations tothe LBT and UBT discussed above, into a (conjectured) complete characterisation.McMullen’s conjecture is now proved, and remains up to the present time perhapsthe most impressive achievement of the combinatorial theory of face numbers:

Theorem 1.4.14 (g-theorem). An integer vector (f0, f1, . . . , fn) is the f -vec-tor of a simple n-polytope if and only if the corresponding sequence (h0, . . . , hn)determined by (1.9) satisfies the following three conditions:

(a) hi = hn−i for i = 0, 1, . . . , n (the Dehn–Sommerville relations);(b) h0 6 h1 6 · · · 6 h[n

2

];(c) h0 = 1, hi+1 − hi 6 (hi − hi−1)

〈i〉 for i = 1, . . . ,[n2

]− 1.

Remark. Condition (b) says that the components of the h-vector ‘grow up tothe middle’, while (c) gives a restriction on the rate of this growth. Both (b) and (c)can be reformulated by saying that the g-vector (g0, . . . , g[n/2]) (see Definition 1.3.1)is an M -vector; this explains the name ‘g-theorem’. The fact that the g-vector isan M -vector implies that the h-vector is also an M -vector (see Exercise 1.4.18).

Both necessity and sufficiency parts of the g-theorem were proved almost si-multaneously (around 1980), although by radically different methods.

The sufficiency part was proved by Billera and Lee [29], [30]. The proof is quiteelementary and relies upon a remarkable combinatorial-geometrical constructioncombining cyclic polytopes (achieving the upper bound for the number of faces)with the operation of ‘adding a pyramid’ (used to produce polytopes achievingthe lower bound). As a result, a simplicial polytope can be produced with anyprescribed g-vector between the minimal and the maximal ones. Another importantconsequence of the results of [222] and [30] is that the GLBC inequalities (1.18) arethe most general linear inequalities satisfied by the f -vectors of simplicial polytopes.


On the other hand, Stanley’s proof [289] of the necessity part of g-theorem(i.e. that the g-vector of a simple polytope is an M -vector) used deep results fromalgebraic geometry, in particular, the Hard Lefschetz theorem for the cohomologyof toric varieties. We shall give this argument in Section 5.3. After the appear-ance of Stanley’s paper combinatorists had been looking for a more elementarycombinatorial proof of his theorem, until in 1993 a first such proof was found byMcMullen [219]. It builds upon the notion of the polytope algebra, which may bethought of as a combinatorial model for the cohomology algebras of toric varieties.Despite being elementary, it was a complicated proof. Later McMullen simplifiedhis approach in [220]. Yet another elementary proof of the g-theorem was given byTimorin [307]. It relies on an interpretation of McMullen’s polytope algebra as thealgebra of differential operators with constant coefficients vanishing on the volumepolynomial of the polytope.

By duality, the UBT and the LBT provide upper and lower bounds for the num-ber of faces of a simple polytope with the given number of facets. Similarly, theg-theorem also provides a characterisation for the f -vectors of simplicial polytopes.During the last three decades some work was done in extending the g-theoremto objects more general than simplicial (or simple) polytopes, although the mostimportant conjecture here remains open since 1971 (see Section 2.5). There arebasically two diverging routes for generalisations of the g-theorem: towards non-polytopal objects (like triangulations of spheres or manifolds), and towards generalconvex polytopes which are neither simple nor simplicial. The former requires ma-chinery from combinatorial topology and commutative algebra; we shall discuss thecorresponding generalisations in more detail in the next chapters. The generalisa-tions of the g-theorem to non-simplicial convex polytopes are mostly beyond thescope of this book; they require algebraic geometry techniques such as intersectionhomology, which we only briefly discuss in Section 5.3.

Exercises.

1.4.15. Prove the following upper bound for the number of k-faces in a simplen-polytope P with f0 vertices:

fk 61

k + 1

(n

k

)f0, for k = 1, . . . , n,

where the equality is achieved only for P = ∆n. Observe that this inequality givesa better upper bound for simple polytopes that the UBT.

1.4.16. Prove Lemma 1.4.7.

1.4.17. Show that the numbers of faces of a stacked n-polytope S with m = f0vertices are given by

fi(S) =(ni

)m−

(n+1i+1

)i for 1 6 i 6 n− 2;

fn−1(S) = (n− 1)m− (n+ 1)(n− 2).

1.4.18. Let h = (h0, h1, . . . , hn) be an integer vector with h0 = 1 and hi = hn−ifor 0 6 i 6 n, and let g = (g0, g1, . . . , g[n/2]) where g0 = 1, gi = hi− hi−1 for i > 0.Show that if g is an M -vector, then h is also an M -vector.

1.4.19. Prove directly that parts (a) and (b) of the g-theorem imply the LBT,while parts (a) and (c) imply the UBT.

Polytopes: additional topics

1.5. Nestohedra and graph-associahedra

Several constructions of series of simple polytopes with remarkable propertiesappeared in the beginning of the 1990s under the common name of ‘generalised as-sociahedra’. (The original associahedron, or Stasheff polytope, was first introducedin homotopy theory [295].) Nowadays generalised associahedra find numerous ap-plications in algebraic geometry [125], the theory of knot and link invariants [38],representation theory and cluster algebras [116], and the theory of operads andgeometric ‘field theories’ originating from quantum physics [296].

Without attempting to overview all aspects of generalised associahedra, wedescribe one particular construction which uses the Minkowski sum and the combi-natorial concept of a building set. Our exposition is much influenced by the originalworks of Feichtner–Sturmfels [110] and Postnikov [265]. The resulting family ofpolytopes is known as nestohedra. Although it does not include all possible gener-alisations of associahedra, this family is wide enough to contain all classical series,and its construction is elementary enough so that it requires no specific knowledge.

The Minkowski sum is a classical geometric construction allowing one to pro-duce new polytopes from known ones, just like the product, hyperplane cut andconnected sum described in Section 1.1. However, a Minkowski sum of simple poly-topes usually fails to be simple. Interesting examples of polytopes can be obtainedby taking Minkowski sums of regular simplices. Simplices in such a Minkowskisum are indexed by a collection S of subsets in a finite set. It was shown in [110]and [265] that a Minkowski sum of regular simplices is a simple polytope if S sat-isfies certain combinatorial condition, identifying it as a building set . The resultingfamily of simple polytopes was called nestohedra in [266] because of its connectionto nested sets considered by De Concini and Procesi [91] in the context of subspacearrangements. An example of a building set is provided by the collection of subsetsof vertices which span connected subgraphs in a given graph. The correspond-ing nestohedra are called graph-associahedra; they were introduced and studied inthe work of Carr and Devadoss [67], and independently in the work of ToledanoLaredo [308] under the name De Concini–Procesi associahedra. These include theclassical series of permutahedra and associahedra.

Minkowski sums of simplices. Recall that the Minkowski sum of two sub-sets A,B ⊂ Rn is defined as

A+B = x + y : x ∈ A, y ∈ B.Proposition 1.5.1. The Minkowski sum of two polytopes is a polytope. More-

over, if P = conv(v1, . . . , vk) and Q = conv(w1, . . . ,wl), then

P +Q = conv(v1 +w1, . . . , vi +wj , . . . , vk +wl).

31

32 POLYTOPES: ADDITIONAL TOPICS

Proof. Follows directly from the definition of Minkowski sum.

For every subset S ⊂ [n+ 1] = 1, . . . , n+ 1 consider the regular simplex

∆S = conv(ei : i ∈ S) ⊂ Rn+1.

Let F be a collection of nonempty subsets of [n+1]. We assume that F containsall singletons i, 1 6 i 6 n+ 1. As usual, denote by |F| the number of elementsin F . Given a subset N ⊂ [n+ 1], denote by F|N the restriction of F to N , i.e.

F|N = S ∈ F : S ⊂ Nof subsets in N . A collection F is connected if [n + 1] cannot be represented as adisjoint union of nonempty subsets N1 and N2 such that for every S ∈ F eitherS ⊂ N1 or S ⊂ N2. Obviously, every collection F splits into the disjoint union ofits connected components.

Remark. In some further considerations we shall allow F to contain somesubsets of [n+ 1] with multiplicities.

Now consider the convex polytope

(1.19) PF =∑

S∈F

∆S ⊂ Rn+1.

The following statement gives some of its basic properties.

Proposition 1.5.2. Let F = F1⊔· · · ⊔Fq be the decomposition into connectedcomponents. Then

(a) PF = PF1 × · · · × PFq;

(b) dimPF = n+ 1− q.Proof. Statement (a) follows from the fact that the polytopes PFi

are con-tained in complementary subspaces for 1 6 i 6 q, so their Minkowski sum is theproduct. Because of (a), it is enough to verify (b) for connected F only. Then weneed to prove that dimPF = n. Observe that PF is contained in the hyperplaneHF =

x ∈ Rn+1 :

∑n+1i=1 xi = |F|

, and therefore dimPF 6 n. If F has a unique

maximal element, then this element is [n+ 1], because F is connected. Hence, PF

has an n-simplex as a Minkowski summand, and therefore dimPF = n. We shallonly need this case in the further considerations, so we skip the rest of the proofand leave it as an exercise.

We now describe two extreme examples of polytopes PF , corresponding to theminimal and the maximal connected collections.

Example 1.5.3 (Simplex). Let S be the collection consisting of all singletonsand the whole set [n+ 1]. Then PS is the regular n-simplex ∆[n+1] shifted by thevector e1 + · · ·+ en+1.

Permutahedron. Let C be the complete collection, consisting of all subsetsin [n+ 1]. The polytope PC is n-dimensional by Proposition 1.5.2.

Theorem 1.5.4. PC can be described as the intersection of the hyperplane

HC =x ∈ Rn+1 :

n+1∑

i=1

xi = 2n+1 − 1

1.5. NESTOHEDRA AND GRAPH-ASSOCIAHEDRA 33

with the halfspaces

HS,> =x ∈ Rn+1 :

∑

i∈S

xi > 2|S| − 1

for all proper subsets S ( [n+ 1]. Moreover, every halfspace above is irredundant,i.e. determines a facet FS of PC, so there are |C| = 2n+1 − 2 facets in total.

Proof. By definition, every point x = (x1, . . . , xn+1) ∈ PC can be written asx =

∑S∈C xS where xS = (xS1 , . . . , x

Sn+1) ∈ ∆S . Then

n+1∑

i=1

xi =∑

S∈C

n+1∑

i=1

xSi =∑

S∈C

1 = |C| = 2n+1 − 1,

which implies that PC ⊂ HC . Similarly,

(1.20)∑

i∈S

xi =∑

T∈C

∑

i∈S

xTi >∑

T⊂S

∑

i∈S

xTi =∑

T⊂S

1 =∣∣C|S

∣∣ = 2|S| − 1,

so PC is contained in all subspaces HS,>.It remains to show that any facet of PC has the form PC ∩ HS , where HS is

the bounding hyperplane for HS,>. Since PC is a Minkowski sum of simplices, eachof its faces G is a Minkowski sum of faces of these simplices. We therefore maywrite G as

∑S∈C∆TS

where TS ⊂ S. By Proposition 1.5.2, if G is a facet (i.e.dimG = n− 1), then the collection T = TS of subsets in [n+ 1] has exactly twoconnected components. (Note that T may contain some subsets more than once.)Let [n+1] = N1⊔N2 and T = T |N1 ⊔T |N2 be the decomposition into components.Then the hyperplane containing the facet G is defined by each of the two equations

(1.21)∑

i∈N1

xi =∣∣T |N1

∣∣ or∑

i∈N2

xi =∣∣T |N2

∣∣.

Since every TS is contained in the corresponding S, we have

(1.22)∣∣T |N1

∣∣ >∣∣C|N1

∣∣ and∣∣T |N2

∣∣ >∣∣C|N2

∣∣.We claim that at least one of these inequalities turns into equality. Indeed, assumethe converse. By (1.20) the minimum of the linear function

∑i∈N1

xi on PC is∣∣C|N1

∣∣, so there is a point x ′ ∈ PC with

(1.23)∑

i∈N1

x′i =∣∣C|N1

∣∣ <∣∣T |N1

∣∣.

Similarly, there is a point x ′′ ∈ PC with∑

i∈N2

x′′i =∣∣C|N2

∣∣ <∣∣T |N2

∣∣.

Since N1 ⊔N2 = [n+ 1], the latter inequality is equivalent to∑

i∈N1x′′i >

∣∣T |N1

∣∣.This together with (1.23) implies that there are points of PC in both open halfspacesdetermined by the first of the equations (1.21), which contradicts the assumptionthat G is a facet. So at least one of (1.22) is an equality, which implies that thehyperplane (1.21) containing G has the form HS (where S is either N1 or N2).

It follows that every facet is contained in the hyperplaneHS for some S. On theother hand, every subset S can be taken as N1 in the construction of the precedingparagraph, which shows that every HS contains a facet.


Having identified the facets, we may derive the following description of thewhole face poset of PC .

Proposition 1.5.5. Faces of PC of dimension k are in one-to-one correspon-dence with ordered partitions of the set [n+ 1] into n+ 1 − k nonempty parts. Aninclusion of faces G ⊂ F occurs whenever the ordered partition corresponding to Gcan be obtained by refining the ordered partition corresponding to F .

Proof. This follows from the fact that two facets FS1 and FS2 of PC havenonempty intersection if and only if S1 ⊂ S2 or S2 ⊂ S1. We skip the details toavoid repetitive arguments; see Theorem 1.5.13 below for a more general result.

Corollary 1.5.6.

(a) PC is a simple polytope;(b) the vertices of PC are obtained by all permutations of the coordinates of

the point (1, 2, 4, . . . , 2n) ∈ Rn+1.

The polytope whose vertices are obtained by permuting the coordinates of agiven point is known as the permutahedron; it has been studied by convex ge-ometers since the beginning of the 20th century. More precisely, given a pointa = (a1, . . . , an+1) ∈ Rn+1 with a1 < a2 < · · · < an+1, define the correspondingpermutahedron as

Pen(a) = conv(σ(a1), . . . , σ(an+1) : σ ∈ Σn+1

),

where Σn+1 denotes the group of permutations of n + 1 elements. In particular,PC = Pen(1, 2, . . . , 2n). All n-dimensional permutahedra Pen(a) are combinatori-ally equivalent; this follows from the following description of their faces:

Theorem 1.5.7.

(a) Every facet of Pen(a) is the intersection of Pen(a) with the hyperplane

HS =x ∈ Rn+1 :

∑

i∈S

xi = a1 + a2 + · · ·+ a|S|

for a proper subset S ( [n+ 1].(b) The faces of Pen(a) are described in the same way as in Proposition 1.5.5.

Proof. This can be proved by mimicking the proof of Theorem 1.5.4. Anotherway to proceed is as follows. Every face of Pen(a) is a set of points where a certainlinear function ϕb = (b, · ) restricted to the polytope achieves its minimum. Wedenote this face by Gb . Then b = (b1, . . . , bn+1) defines an ordered partition[n+ 1] = N1 ⊔ · · · ⊔Nk according to the sets of equal coordinates of b. Namely, ifbk = bl then k and l are in the same Ni, while if bk < bl then k ∈ Ni and l ∈ Nj withi < j. It can be shown that (a) dimGb = n+1−k, and (b) the face Gb only dependson the ordered partition above and does not depend on the particular values of bk.In particular, the vertices of Pen(a) correspond to ϕb where all coordinates of b

are different, while the facets correspond to ϕb where all coordinates of b are either0 or 1. We leave the details to the reader.

We shall denote an n-dimensional combinatorial permutahedron by Pen. Theclassical permutahedron corresponds to a = (1, 2, . . . , n+1). It can also be obtainedas a Minkowski sum (1.19) as follows (we leave the proof as an exercise):


Proposition 1.5.8. Let I be the collection of all subsets of cardinality 6 2in [n+ 1]. Then

PI = Pen(1, 2, . . . , n+ 1).

The polytope PI is the Minkowski sum of segments of the form conv(ei, ej)for 1 6 i < j 6 n + 1, shifted by the vector e1 + · · · + en+1. Minkowski sums ofsegments are known as zonotopes ; this family of polytopes has many remarkableproperties [325, §7.3]. However, general zonotopes are rarely simple; the permuta-hedron is one of the few exceptions.

Building sets and nestohedra. We now return to general Minkowskisums (1.19). The next statement gives a description of PF in terms of inequalities,and generalises the first part of Theorem 1.5.4.

Proposition 1.5.9 ([110, Proposition 3.12]). PF can be described as the in-tersection of the hyperplane

HF =x ∈ Rn+1 :

n+1∑

i=1

xi = |F|

with the halfspaces

HT,> =x ∈ Rn+1 :

∑

i∈T

xi >∣∣F|T

∣∣

corresponding to all proper subsets T ( [n+ 1].

Proof. Since PF is a Minkowski summand in the permutahedron PC , it isdefined by inequalities of the form

∑i∈T xi > bT for some parameters bT . The

minimum value of the linear function∑i∈T xi on a simplex ∆S equals one if S ⊂ T ,

and equals zero otherwise. Therefore, bT =∣∣F|T

∣∣, as needed.

It follows that PF can be obtained by iteratively cutting the n-simplex

HF ∩ x : xi > 1 for i = 1, . . . , n+ 1by the hyperplanes HT,> corresponding to subsets T of cardinality > 2. In the caseof the permutahedron, each of these cuts is nontrivial, that is, the correspondinghyperplane is not redundant. In general, the description of PF in Proposition 1.5.9is redundant. The concept of a building set will allow us to achieve an irredundantdescription of PF for certain F and therefore describe the face posets. The resultingpolytopes PF will be simple; furthermore they will be obtained from a simplex bya sequence of face truncations.

Definition 1.5.10. A collection B of nonempty subsets of [n + 1] is called abuilding set on [n+ 1] if the following two conditions are satisfied:

(a) S′, S′′ ∈ B with S′ ∩ S′′ 6= ∅ implies S′ ∪ S′′ ∈ B;(b) i ∈ B for all i ∈ [n+ 1].

Remark. The terminology comes from a more general notion of a building setin a finite lattice, so that the building set above corresponds to the case of theBoolean lattice 2[n+1]. We do not give the general definition because it requires toomuch poset terminology. See [110, §3] for the details.


Note that a building set B on [n + 1] is connected if and only if [n + 1] ∈ B.Given S ( [n+ 1] define the contraction of S from B as

B/S = T \ S : T ∈ B, T \ S 6= ∅ = S′ ⊂ [n+ 1]\S : S′ ∈ B or S′ ∪ S ∈ B.The restriction B|S and the contraction B/S are building sets on S and [n+ 1] \ Srespectively. Note that B|S is connected if and only if S ∈ B. If B is connected,then B/S is also connected for any S.

We now consider polytopes PB (1.19) corresponding to building sets B. Thefollowing specification of Proposition 1.5.9 gives an irredundant description of PB

as an intersection of halfspaces.

Proposition 1.5.11 ([110], [265]). We have

PB =x ∈ Rn+1 :

n+1∑

i=1

xi = |B|,∑

i∈S

xi >∣∣B|S

∣∣ for every S ∈ B.

If B is connected, then this representation is irredundant, that is, every hyperplaneHS =

x ∈ Rn+1 :

∑i∈S xi =

∣∣B|S∣∣ with S 6= [n+ 1] defines a facet FS of PB (so

that the number of facets of PB is |B| − 1).

Proof. The halfspace HT,> in the presentation of PB from Proposition 1.5.9is irredundant if the intersection of PB with the corresponding hyperplane HT is afacet. This intersection is a face of PB given by

PB|T + PB/T

(since PB|T and PB/T lie in complementary subspaces, their Minkowski sum is actu-ally a product). In order for this face to have codimension one in PB, it is necessary,by Proposition 1.5.2, that the collection B|T is connected. This condition is equiva-lent to T ∈ B. If B is connected, then this condition is also sufficient, because thenB/T is also connected, and dim(PB|T + PB/T ) = n− 1 by Proposition 1.5.2.

Corollary 1.5.12. If B is a connected building set on [n+1], then every facetof PB can be written as

PB|T × PB/T

for some T ∈ B \ [n+ 1].

Theorem 1.5.13. The intersection of facets FS1∩ · · · ∩ FSkis nonempty (and

therefore is a face of PB) if and only if the following two conditions are satisfied:

(a) for any i, j, 1 6 i < j 6 k, either Si ⊂ Sj, or Sj ⊂ Si, or Si ∩ Sj = ∅;(b) if the sets Si1 , . . . , Sip are pairwise nonintersecting, then Si1⊔· · ·⊔Sip /∈ B.

Definition 1.5.14. A subcollection S1, . . . , Sk ⊂ B satisfying conditions (a)and (b) of Theorem 1.5.13 is called a nested set. Following [266], we refer topolytopes PB (1.19) corresponding to building sets B as nestohedra.

Proof of Theorem 1.5.13. Assume FS1∩ · · · ∩ FSk6= ∅.

If (a) fails, then Si ∪ Sj ∈ B, and for any x ∈ FSi∩ FSj

we have∑

k∈Si

xk =∣∣B|Si

∣∣,∑

k∈Sj

xk =∣∣B|Sj

∣∣,∑

k∈Si∪Sj

xk >∣∣B|Si∪Sj

∣∣.

Adding the first two equalities and subtracting the third inequality we obtain∑

k∈Si∩Sj

xk 6∣∣B|Si

∣∣+∣∣B|Sj

∣∣−∣∣B|Si∪Sj

∣∣ <∣∣B|Si

∣∣+∣∣B|Sj

∣∣−∣∣B|Si

∪B|Sj

∣∣ =∣∣B|Si∩Sj

∣∣


where the second inequality is strict because B|Si∪ B|Sj

( B|Si∪Sj. Now the

inequality∑k∈Si∩Sj

xk <∣∣B|Si∩Sj

∣∣ contradicts Proposition 1.5.9.

If (b) fails, then Si1 ⊔ · · · ⊔ Sip ∈ B, and for any x ∈ FSi1∩ · · · ∩ FSip

we have∑

k∈Siq

xk =∣∣B|Siq

∣∣ for 1 6 q 6 p, and∑

k∈Si1⊔···⊔Sip

xk >∣∣B|Si1⊔···⊔Sip

∣∣.

Subtracting the first p equalities from the last inequality we obtain∣∣B|Si1

∣∣+ · · ·+∣∣B|Sip

∣∣ >∣∣B|Si1⊔···⊔Sip

∣∣.This leads to a contradiction because Si1 ⊔ · · · ⊔ Sip ∈ B.

Now assume that both (a) and (b) are satisfied. We need to show that FS1 ∩· · · ∩ FSk

6= ∅.We write x =

∑T∈B x

T and note that the inequality∑i∈S xi >

∣∣B|S∣∣ defining

the facet FS turns into equality if and only if xTi = 0 for every T ∈ B, T 6⊂ S andi ∈ S (this follows from (1.20)). Hence,

(1.24) x =∑

T∈B

xT ∈ FS1∩ · · · ∩ FSk

⇐⇒ xTi = 0 whenever T ∈ B, T 6⊂ Sj , i ∈ Sj , for 1 6 j 6 k.

We therefore need to find x whose coordinates satisfy the k conditions on the righthand side of (1.24). Given T ∈ B, the jth condition is not void only if T 6⊂ Sjand T ∩ Sj 6= ∅. We may assume without the loss of generality that the firstk′ conditions in (1.24) are not void, and the rest are void. That is, T 6⊂ Sj andT ∩ Sj 6= ∅ for 1 6 j 6 k′, while T ⊂ Sj or T ∩ Sj = ∅ for j > k′. Then weclaim that T \ (S1 ∪ · · · ∪ Sk′ ) 6= ∅. Indeed, otherwise choosing among S1, . . . , Sk′the maximal subsets Si1 , . . . , Sip (which are pairwise disjoint by (a)) we obtain

Si1 ⊔ · · · ⊔Sip = T ∪Si1 ∪ · · · ∪ Sip ∈ B, which contradicts (b). Now setting xTi = 1

for only one i ∈ T \ (S1 ∪ · · · ∪ Sk′) and xTi = 0 for the rest, we obtain the requiredpoint x in the intersection FS1∩ · · · ∩ FSk

.

From the description of the face lattice of nestohedra in Theorem 1.5.13 it iseasy to deduce their following main property.

Theorem 1.5.15. Every nestohedron PB is a simple polytope.

Proof. By Proposition 1.5.2 we may assume that B is connected. A collectionS1, . . . , Sk may satisfy both conditions of Theorem 1.5.13 only if k 6 n.

Example 1.5.16. If B is a connected building set on a 2-element set, then PB

is an interval I1. If B is a connected building set on a 3-element set, then PB is apolygon, and only m-gons with 3 6 m 6 6 arise in this way.

More examples will appear in the next subsections.Proposition 1.5.11 gives a particular way to obtain a nestohedron PB from

a simplex by a sequence of hyperplane cuts. The next result shows that thesehyperplane cuts can be organised in such a way that we get a sequence of facetruncations (see Construction 1.1.12).

Let B0 ⊂ B1 be building sets on [n+1], and S ∈ B1. We define a decompositionof S into elements of B0 as S = S1⊔· · ·⊔Sk, where Sj are pairwise nonintersectingelements of B0 and k is minimal among such disjoint representations of S. It canbe easily seen that this decomposition exists and is unique.


Lemma 1.5.17. Let B0 ⊂ B1 be connected building sets on [n+1]. Then PB1 is

obtained from PB0 by a sequence of truncations at faces Gi =⋂kij=1 FSi

jcorrespond-

ing to the decompositions Si = Si1 ⊔ · · · ⊔Siki of elements Si ∈ B1 \B0, numbered in

any order that is reverse to inclusion (i.e. Si ⊇ Si′ ⇒ i 6 i′).

Proof. We use induction on the number N = |B1|− |B0|. For N = 1, we haveB1 = B0∪S1. We shall show that PB1 is obtained from PB0 by a single truncationat the face G = FS1

1∩· · ·∩FS1

k, where S1 = S1

1 ⊔· · ·⊔S1k is the decomposition of S1

into elements of B0. Let PB0 denote the polytope obtained by truncating PB0 at G.

Since both PB0 and PB1 are n-dimensional polytopes (here we use the assumptionthat both B0 and B1 are connected), it is enough to verify that the face poset of

PB1 is a subposet of the face poset of PB0 (see Exercise 1.1.20).The facets of PB1 are FS1 and FSj

with Sj ∈ B0. We first consider a nonemptyintersection of the form FS1 ∩ . . . ∩ FSℓ

in PB1 , i.e. a nested set S1, . . . , Sℓ of B1,with all Sj ∈ B0. Then obviously, S1, . . . , Sℓ is a nested set of B0, i.e. FS1 ∩ . . .∩FSℓ6= ∅ in PB0 . Furthermore, since S1 is the only element of B1 \B0, we have that

Sj1 ⊔ · · · ⊔ Sjp 6= S1 = S11 ⊔ · · · ⊔ S1

k

for any j1, . . . , jp ⊂ [ℓ]. The latter condition implies that S11 , . . . , S

1k 6⊂

S1, . . . , Sℓ, i.e. FS1 ∩ . . . ∩ FSℓ6⊂ G in the face poset of PB0 . By the de-

scription of the face poset of PB0 given in Construction 1.1.12, this implies that

FS1 ∩ . . . ∩ FSℓ6= ∅ in PB0 .

Now we consider a nonempty intersection of the form FS1 ∩ FS1 ∩ . . . ∩ FSℓin

PB1 , i.e. a nested set S1, S1, . . . , Sℓ of B1, with Sj ∈ B0 and S1 ∈ B1 \ B0. Weclaim that S1

1 , . . . , S1k, S1, . . . , Sℓ is a nested set of B0, i.e. G∩FS1 ∩ . . .∩FSℓ

6= ∅in PB0 . To do this we need to verify (a) and (b) of Theorem 1.5.13.

We need to check condition (a) for pairs of the form S1p , Sq; for other pairs it

is obvious. That is, we need to check that if S1p ∩ Sq 6= ∅, then one of S1

p , Sq is

contained in the other. The condition S1p ∩Sq 6= ∅ implies that S1 ∩Sq 6= ∅. Since

S1, S1, . . . , Sℓ is a nested set of B1, we obtain that S1p ⊂ S1 ⊂ Sq or Sq ⊂ S1.

By the minimality of the decomposition S1 = S11 ⊔ · · · ⊔ S1

k, the inclusion Sq ⊂ S1

implies that Sq is contained in some S1r , which can be only S1

p , since S1p ∩ Sq 6= ∅.

To verify condition (b) of Theorem 1.5.13 for S11 , . . . , S

1k, S1, . . . , Sℓ, we con-

sider a subcollection S1i1, . . . , S1

ip, Sj1 , . . . , Sjq consisting of pairwise nonintersect-

ing subsets. We need to check that its union is not in B0. For obvious reasons, wemay assume that p > 0 and q > 0. Since S1, S1, . . . , Sℓ is a nested set of B1,we have that either Sji ⊂ S1 or Sji ∩ S1 = ∅ for each i = 1, . . . , q. Suppose thatS1i1⊔· · · ⊔S1

ip⊔Sj1 ⊔· · · ⊔Sjq ∈ B0. Then S1⊔Sj1 ⊔· · · ⊔Sjq ∈ B1 by the definition

of the building set. If any of Sji is disjoint with S1, then we get a contradictionwith condition (b) for the nested set S1, S1, . . . , Sℓ of B1. Therefore, Sji ⊂ S1 fori = 1, . . . , q, so that S1

i1⊔· · ·⊔S1

ip⊔Sj1⊔· · ·⊔Sjq ⊂ S1. By the argument of the previ-

ous paragraph, for each Sji we have that Sji ⊂ S1r or S1

r ⊂ Sji for some r = 1, . . . , k.Then it follows from the minimality of the decomposition S1 = S1

1 ⊔ · · · ⊔ S1k and

the definition of a building set that S1i1⊔ · · · ⊔ S1

ip⊔ Sj1 ⊔ · · · ⊔ Sjq = S1, which

contradicts the assumption that S1 /∈ B0.Hence, S1

1 , . . . , S1k, S1, . . . , Sℓ is a nested set of B0. Similarly to the case

considered in the previous paragraph, we also obtain that FS1 ∩ . . . ∩ FSℓ6⊂ G in


the face poset of PB0 . Once again, by the description of the face poset of PB0 given

in Construction 1.1.12, this implies that FS1 ∩ FS1 ∩ . . . ∩ FSℓ6= ∅ in PB0 .

It follows that the face poset of PB1 is indeed contained as a subposet in the

face poset of PB0 , and thus PB1 = PB0 .It now remains to finish the induction. Assuming the theorem holds forM < N ,

we shall prove it forM = N . Since S1 is not contained in any other Si, the collectionof sets B′

0 = B0 ∪ S1 is a building set. By the induction assumption, PB′0

isobtained from PB0 by truncation at the face corresponding to the decompositionof S1, and PB1 is obtained from PB′

0by a sequence of truncations corresponding to

the decompositions of Si for i = 2, . . . , N .

Remark. The proof given above only establishes a combinatorial equivalence

between PB1 and PB0 . Since Proposition 1.5.11 gives a geometric presentationof nestohedra by a sequence of hyperplane cuts, it follows easily that the facetruncations of PB0 giving PB1 are also geometric.

Theorem 1.5.18. Every nestohedron PB corresponding to a connected buildingset B can be obtained from a simplex by a sequence of face truncations.

Proof. Assume that B is a connected building set on [n + 1]. Then we haveS ⊂ B, where S is the connected building set of Example 1.5.3, whose correspondingnestohedron is an n-simplex. Now apply Lemma 1.5.17.

The following construction, suggested by N. Erokhovets, will allow us to showthat every nestohedron can be obtained from a connected building set, up to com-binatorial equivalence.

Construction 1.5.19 (Substitution of building sets). Let B1, . . . ,Bn+1 beconnected building sets on [k1], . . . , [kn+1]. Then, for every connected building set Bon [n+1], we define a connected building set B(B1, . . . ,Bn+1) on [k1]⊔. . .⊔[kn+1] =[k1 + . . .+ kn+1], consisting of elements Si ∈ Bi and

⊔i∈S

[ki], where S ∈ B.

When B1, . . . ,Bn are singletons 1, . . . , n, we shall write B(1, 2, . . . , n,Bn+1)instead of B

(1, 2, . . . , n,Bn+1

).

Lemma 1.5.20. Let B, B1, . . . ,Bn+1 be connected building sets on [n + 1],[k1], . . . , [kn+1], and let B′ = B(B1, . . . ,Bn+1). Then PB′ ≈ PB×PB1 × · · ·×PBn+1.

Proof. Set B′′ = B ⊔ B1 ⊔ · · · ⊔ Bn+1 and define the map ϕ : B′′ → B′ by

ϕ(S) =

S if S ∈ Bi⊔i∈S

[ki] if S ∈ B .

Then ϕ generates a bijection between B′′ \ B′′max and B′ \ [k1 + · · ·+ kn+1], where

B′′max = [n+ 1], [k1], . . . , [kn+1] denotes the collection of maximal sets of B′′. Let

S ⊂ B \ [n + 1] and Si ⊂ Bi \ [ki]. Notice that ϕ(S) ∪⋃n+1i=1 ϕ(Si) is a nested set

of B′ if and only if S is a nested set of B and Si is a nested set of Bi for all i. Itfollows that PB′ ≈ PB′′ = PB × PB1 × · · · × PBn+1 .

Example 1.5.21. Assume that each of B, B1, B2 is the building set1, 2, 1, 2 corresponding to the segment I. Let us describe the build-ing set B(B1,B2). In the building set a, b, a, b, we substitute a byB1 = 1, 2, 1, 2 and b by B2 = 3, 4, 3, 4. As a result, we obtain the


connected building set B′, consisting of 1, 2, 3, 4, 1, 2, 3, 4, [4]. Its cor-responding nestohedron is obtained by truncating a 3-simplex at two nonadjacentedges; it is combinatorially equivalent to a 3-cube.

The facet correspondence ϕ between the combinatorial cubes PB × PB1 × PB2

and PB′ is given by

1 ∈ B1 7→ 1 ∈ B′, 2 ∈ B1 7→ 2 ∈ B′,

3 ∈ B2 7→ 3 ∈ B′, 4 ∈ B2 7→ 4 ∈ B′,

a ∈ B2 7→ 1, 2 ∈ B′, b ∈ B2 7→ 3, 4 ∈ B′.

Example 1.5.22. Let B = 1, . . . , n+1, [n+1] be the building set corre-sponding to the simplex ∆n and let B1, . . . ,Bn+1 be arbitrary connected buildingsets on [k1], . . . , [kn+1]. Then

B′ = B(B1, . . . ,Bn+1) = (B1 ⊔ · · · ⊔ Bn+1) ∪ [k1 + · · ·+ kn+1],

and PB′ ≈ ∆n × PB1 × · · · × PBn+1 .

Proposition 1.5.23. For each nestohedron PB there exists a connected buildingset B′ such that PB ≈ PB′ .

Proof. Indeed, any building set B can be written as B1 ⊔ · · · ⊔ Bk, where Biare connected building sets on [ki+1]. Define a building set B = B1(1, . . . , k1,B2)⊔B3⊔· · ·⊔Bk, giving the same combinatorial polytope. We have that B is a product(disjoint union) of k−1 connected building sets. Then we apply again a substitution

to B, and so on. At the end we obtain a connected building set B′.

Graph associahedra.

Definition 1.5.24. Let Γ be a graph on the vertex set [n + 1] without loopsand multiple edges (a simple graph). The graphical building set BΓ consists of allnonempty subsets S ⊂ [n+ 1] such that the graph Γ|S is connected.

The nestohedron PΓ = PBΓ corresponding to a graphical building set is calleda graph-associahedron [67].

Example 1.5.25 (associahedron). Let Γ be a ‘path’ with n edges i, i+ 1 for1 6 i 6 n. Then BΓ consists of all ‘segments’ of the form [i, j] = i, i+ 1, . . . , jwhere 1 6 i 6 j 6 n+ 1. To describe the face poset of the corresponding polytopePΓ it is convenient to use brackets in a string of n + 2 letters a1a2 · · ·an+2. Weassociate with a segment [i, j] a pair of brackets before ai and after aj+1. Using The-orem 1.5.13 it is easy to see that the facets corresponding to n different segmentsintersect at a vertex if and only if the corresponding bracketing of a1a2 · · ·an+2

with n pairs of brackets is correct. In particular, the vertices of BΓ correspondto all different ways to obtain a product a1a2 · · · an+2 when multiplication is notassociative. The number of vertices is therefore equal to 1

n+2

(2n+2n+1

), the (n + 1)th

Catalan number. Two vertices are adjacent if and only if the bracketing correspond-ing to one vertex can be obtained from the bracketing corresponding to the othervertex by deleting a pair of brackets and inserting, in a unique way, another pair ofbrackets different from the deleted one. That is, two vertices are adjacent if theycorrespond to a single application of the associative law. This explains the nameassociahedron for the polytope PΓ of this example; we shall denote it Asn.


Figure 1.4. 3-dimensional associahedron and the corresponding graph.

Proposition 1.5.11 describes the associahedron as the result of consecutive hy-perplane cuts of a simplex, see Figure 1.4 for the case n = 3. It can also be obtainedby hyperplane cuts from a cube, as described in the next theorem.

Theorem 1.5.26. The image of Asn under a certain affine transformationRn+1 → Rn is the intersection of the cube

y ∈ Rn : 0 6 yj 6 j(n+ 1− j) for 1 6 j 6 n

with the halfspaces y ∈ Rn : yj − yk + (j − k)k > 0

for 1 6 k < j 6 n.

Proof. Proposition 1.5.11 gives the following presentation:

(1.25) Asn =x ∈ Rn+1 :

n+1∑

k=1

xk =(n+ 1)(n+ 2)

2,

j∑

k=i

xk >(j − i+ 1)(j − i+ 2)

2for 1 6 i 6 j 6 n+ 1

.

Apply the affine transformation Rn+1 → Rn given by

(x1, . . . , xn+1) 7→ (z1, . . . , zn) where zl =

l∑

k=1

xk, for 1 6 l 6 n.

Now we rewrite the inequalities of (1.25) in the new coordinates (z1, . . . , zn). Theinequalities with i = 1 (corresponding to the facets FS with 1 ∈ S) become

(1.26) zj >j(j + 1)

2, for 1 6 j 6 n.

Inequalities (1.25) with j = n+ 1 (corresponding to FS with n+ 1 ∈ S) become

(n+ 1)(n+ 2)

2− zi−1 >

(n+ 2− i)(n+ 3− i)2

, for 2 6 i 6 n+ 1,

or equivalently,

(1.27) zj 6 (n+ 2)j − j(j + 1)

2, for 1 6 j 6 n.


The remaining inequalities (1.25) take the form

(1.28) zj − zi−1 >(j − i+ 1)(j − i+ 2)

2for 1 < i 6 j < n+ 1.

Now the required presentation is obtained from (1.26), (1.27) and (1.28) by applying

the shift yj = zj − j(j+1)2 and setting k = i− 1.

Example 1.5.27. The case n = 3 of Theorem 1.5.26 is shown in Figure 1.5(right). We start with a 3-cube (more precisely, a parallelepiped) given by the

F1

F2F3

F4

F1,2

F2,3

F3,4

F1,2,3

F2,3,4

Figure 1.5. 3-dimensional associahedron cut from a cube.

inequalities0 6 y1 6 3, 0 6 y2 6 4, 0 6 y3 6 3,

and cut it by the three hyperplanes

y2 − y1 + 1 = 0, y3 − y1 + 2 = 0, y3 − y2 + 2 = 0.

Another way to cut a 3-dimensional combinatorial associahedron from a 3-cubeis shown in Figure 1.5 (left); this time we cut three nonadjacent and pairwiseorthogonal edges. The two associahedra in Figure 1.5 are not affinely equivalent.

The associahedron Asn first appeared (as a combinatorial object) in the work ofStasheff [295] as the space of parameters for the higher associativity of the (n+2)-fold product map in an H-space. For more information about the associahedra werefer to [125] and [49, Lec. II], where the reader may find other geometric andcombinatorial realisations of Asn.

Example 1.5.28 (permutahedron revisited). Let Γ be a complete graph; thenBΓ is the complete building set C and PΓ is the permutahedron Pen, see Figure 1.6for the case n = 3.

Example 1.5.29 (cyclohedron). Let Γ be a ‘cycle’ consisting of n + 1 edgesi, i + 1 for 1 6 i 6 n and n + 1, 1. The corresponding PΓ is known as thecyclohedron Cyn, or Bott–Taubes polytope, see Figure 1.7. It was first introducedin [38] in connection with the study of link invariants.

Example 1.5.30 (stellahedron). Let Γ be a ‘star’ consisting of n edgesi, n+ 1, 1 6 i 6 n, emanating from the point n + 1. The corresponding PΓ

is known as the stellahedron Stn, see Figure 1.8.


Figure 1.6. 3-dimensional permutahedron and the corresponding graph.

Figure 1.7. 3-dimensional cyclohedron and the corresponding graph.

Figure 1.8. 3-dimensional stellahedron and the corresponding graph.

Exercises.


1.5.31. Every collection F of subsets in [n+ 1] may be completed in a uniqueway to a building set by iteratively adding to F the unions S1 ∪ S2 of intersecting

sets (S1 ∩ S2 6= ∅) until the process stops. Denote the resulting building set by F .

Show that F is connected if and only if F is connected, and that dimPF = dimPF

(hint: for a pair of intersecting sets S1, S2 compare the polytopes ∆S1 +∆S2 and∆S1 +∆S2 +∆S1∪S2). Use this fact to complete the proof of Proposition 1.5.2.

1.5.32. Finish the argument in the proof of Proposition 1.5.5.

1.5.33. Complete the details in the proof of Theorem 1.5.7.

1.5.34. Prove Proposition 1.5.8.

1.5.35. Given two building sets B0 ⊂ B1, show that a decomposition S =S1 ⊔ · · · ⊔ Sk of S ∈ B1 into elements Si ∈ B0 with minimal k is unique.

1.5.36. A combinatorial polytope obtained from a simplex by a sequence offace truncations is called a truncated simplex. Notice that a truncated simplexis a simple polytope. By Theorem 1.5.18, every nestohedron corresponding to aconnected nested set is a truncated simplex. Give an example of

(a) a simple polytope which is not a truncated simplex;(b) a truncated simplex which is not a nestohedron;(c) a Minkowski sum of simplices PF given by (1.19) which is not a truncated

simplex.

1.5.37. Consider the following connected building set on [4]:

B =1, 2, 3, 4, 1, 2, 3, 2, 3, 4, [4]

.

Then PB is combinatorially equivalent to the polytope shown in Fig. 1.2, right.

1.6. Flagtopes and truncated cubes

Definition 1.6.1. A simple polytope P is called a flagtope (or flag polytope)if every collection of its pairwise intersecting facets has a nonempty intersection.(The origins of this terminology will be explained in Section 2.3.)

Flagtopes and flag simplicial complexes (which are the subject of Section 2.3)feature in the well-known Charney–Davis conjecture [69] and its generalisation dueto Gal [123]. According to the Gal conjecture, the components of the γ-vector ofa flagtope are nonnegative.

Gal’s conjecture is important as it connects the combinatorial study of poly-topes and sphere triangulations to differential geometry and topology of manifolds.This conjecture has been proved in special cases.

Although not all nestohedra are flagtopes (a simplex is the easiest counterex-ample), flag nestohedra constitute an important family. In particular, all graph-associahedra (and therefore, the classical series of associahedra and permutahedra)are flagtopes, see Proposition 1.6.6.

As we have seen in Theorem 1.5.26, the associahedron can be obtained byhyperplane cuts from a cube. In this section we give a proof of a much more generaland precise result: a nestohedron is a flagtope if and only if it can be obtained froma cube by a sequence of truncations at faces of codimension 2 (we refer to suchpolytopes as 2-truncated cubes), see Theorem 1.6.16. This result was proved byBuchstaber and Volodin in [64, Theorem 6.6]. On the other hand, it can be easily

1.6. FLAGTOPES AND TRUNCATED CUBES 45

seen that the Gal conjecture is valid for 2-truncated cubes (see Proposition 1.6.12).This observation (formulated in terms of the dual operation of stellar subdivisionat an edge) was present in the work of Charney–Davis and later Gal, and used tosupport their conjectures. As a corollary we obtain that the Gal conjecture holdsfor all flag nestohedra.

Example 1.6.2.1. The cube In is a flagtope, but the simplex ∆n is not if n > 1.2. The product P ×Q of two flagtopes is a flagtope.3. The connected sum P # Q of two simple n-polytopes and the vertex trun-

cation vt(P ) are not flagtopes if n > 1.

A polytope P is said to be triangle-free if it does not contain a triangular 2-face.

Proposition 1.6.3. A flagtope is triangle-free.

Proof. Assume that a flagtope P of dimension n contains a triangular 2-face Twith vertices v1, v2, v3 and edges e1, e2, e3, where vi is opposite to ei. Since P issimple, each ei is an intersection of n−1 facets. Hence, for each i = 1, 2, 3 there is aunique facet, say Fi, which contains the edge ei but not the triangle T . Also, T is an

intersection of n−2 facets, and we may assume that T =⋂n+1i=4 Fi. Now we observe

that⋂n+1i=1,i6=j Fi = vj for j = 1, 2, 3. This implies that the intersection of any pair of

facets among F1, . . . , Fn+1 is nonempty. On the other hand,⋂n+1i=1 Fi = ∅ because

P is simple and n-dimensional. Therefore P cannot be a flagtope.

The converse to Proposition 1.6.3 does not hold in general (see exercises), butit is valid for polytopes with few facets:

Theorem 1.6.4 ([33]). If P is a triangle-free convex n-polytope then fi(P ) >fi(I

n) for i = 0, . . . , n. In particular, such P has at least 2n facets. Furthermore ifP is simple then

(a) fn−1(P ) = 2n implies that P = In;(b) fn−1(P ) = 2n+ 1 implies that P = P5 × In−2 where P5 is a pentagon;(c) fn−1(P ) = 2n + 2 implies that P = P6 × In−2 or P = Q × In−3 or

P = P5×P5× In−4 where P6 is a hexagon and Q is a 3-polytope obtainedby truncating a pentagonal prism at one of its edges forming a pentagonalfacet.

Corollary 1.6.5. A triangle-free simple n-polytope with at most 2n+2 facetsis a flagtope.

Another source of examples of flagtopes is provided by graph-associahedra (seeDefinition 1.5.24):

Proposition 1.6.6. Every graph-associahedron PΓ is a flagtope.

Proof. Let FS1 , . . . , FSkbe a set of facets of PΓ with nonempty pairwise in-

tersections. We need to check that FS1 ∩ · · · ∩ FSk6= ∅, i.e. that condition (b) of

Theorem 1.5.13 is satisfied (condition (a) holds automatically as it depends onlyon pairwise intersections). Let Si1 , . . . , Sip be pairwise nonintersecting sets amongS1, . . . , Sk; then Sir ∪ Sis /∈ BΓ for 1 6 r < s 6 p because FSir

∩ FSis6= ∅. There-

fore, all subgraphs Γ|Sir∪Sisare disconnected, which implies that Γ|Si1∪···∪Sip

is

also disconnected. Thus, Si1 ∪ · · · ∪ Sip /∈ BΓ, and FS1 ∩ · · · ∩ FSk6= ∅ by Theo-

rem 1.5.13.


The main conjecture about the face numbers of flagtopes uses the notion of theγ-vector γ(P ) = (γ0, . . . , γ[n/2]) (see Definition 1.3.10):

Conjecture 1.6.7 (Gal [123]). A flagtope P of dimension n satisfies γi(P ) >0 for i = 0, . . . ,

[n2

].

We shall verify the Gal conjecture for all flag nestohedra. To do this we shallshow that any flag nestohedron can be obtained by consecutively truncating a cubeat codimension-2 faces.

Our first task is therefore to describe how the γ-vector changes under facetruncations. For this it is convenient to use the H-polynomial given by (1.9), andthe γ-polynomial

γ(P )(τ) = γ0 + γ1τ + · · ·+ γ[n/2]τ[n/2].

Proposition 1.6.8. Let Q be the polytope obtained by truncating a simple n-polytope P at a k-dimensional face G. Then

(a) H(Q) = H(P )(s, t) + stH(G)H(∆n−k−2),

(b) γ(Q) = γ(P ) + τγ(G)γ(∆n−k−2).

Proof. The truncation removes G and creates a face G×∆n−k−1, so that

fi(Q) = fi(P ) + fi(G×∆n−k−1)− fi(G), for 0 6 i 6 n.

Hence, F (Q) = F (P ) + tF (G)F (∆n−k−1)− tn−kF (G), and

H(Q) = H(P ) + tH(G)H(∆n−k−1)− tn−kH(G)

= H(P ) + tH(G)

(n−k−1∑

i=0

sitn−k−1−i − tn−k−1

)

= H(P ) + stH(G)

n−k−2∑

j=0

sjtn−k−2−j

= H(P ) + stH(G)H(∆n−k−2) ,

which proves (a). Furthermore,

[n2 ]∑

i=0

γi(Q)(st)i(s+ t)n−2i = H(Q) =

[n2 ]∑

i=0

γi(P )(st)i(s+ t)n−2i

+ st

[ k2 ]∑

p=0

γp(G)(st)p(s+ t)k−2p

[n−k−22 ]∑

q=0

γq(∆n−k−2)(st)q(s+ t)n−k−2−2q

=

[n2 ]∑

i=0

γi(P )(st)i(s+t)n−2i+

[ k2 ]∑

p=0

[n−k−22 ]∑

p=0

γp(G)γq(∆n−k−2)(st)p+q+1(s+t)n−2(p+q+1).

Hence, γi(Q) = γi(P ) +∑

p+q=i−1 γp(G)γq(∆n−k−2), which proves (b).

Definition 1.6.9. We refer to a truncation at a codimension-2 face as a 2-truncation. A combinatorial polytope obtained from a cube by 2-truncations willbe called a 2-truncated cube.

The following corollary is the dual of a result from [123]:


Corollary 1.6.10. Let the polytope Q be obtained from a simple polytope Pby 2-truncation at a face G. Then

(a) H(Q) = H(P ) + stH(G),(b) γ(Q) = γ(P ) + τγ(G).

Proposition 1.6.11. Each face of a 2-truncated cube is a 2-truncated cube.

Proof. It is enough to show that if P is a 2-truncated cube, then all thefacets of P are 2-truncated cubes. The proof is by induction on the number offace truncations. Let the polytope Q be obtained from a 2-truncated cube P by2-truncation at a face G of codimension 2. Then the new facet is G × I, and itis a 2-truncated cube by the induction assumption. Every other facet F ′ of thepolytope Q is either a facet of P , or obtained from a facet F ′′ of P by 2-truncationat a face G′ ⊂ F ′′.

Proposition 1.6.12. Any 2-truncated cube P satisfies γi(P ) ≥ 0, i.e. the Galconjecture holds for 2-truncated cubes.

Proof. We proceed by induction on the dimension of P , using Proposi-tion 1.6.11 and the formula γ(Q) = γ(P ) + τγ(G).

Here is a criterion for a nestohedron to be a flagtope.

Proposition 1.6.13 ([64], [266]). Let B be a building set on [n+1]. Then thenestohedron PB is a flagtope if and only if for every element S ∈ B with |S| > 1there exist elements S′, S′′ ∈ B such that S′ ⊔ S′′ = S.

Proof. Suppose PB is a flagtope. Consider an element S ∈ B. Then wemay write S = S1 ⊔ · · · ⊔ Sk, where S1, . . . , Sk ∈ B \ S and k is minimal amongsuch decompositions of S. Then for any subset J ⊂ [k] with 1 < |J | < k wehave that

⊔j∈J Sj /∈ B, since otherwise k can be decreased. If k > 2 then, by

Theorem 1.5.13, the facets FS1 , . . . , FSkof PB intersect pairwise, but have empty

common intersection. Therefore, k = 2.Suppose for each element S ∈ B with |S| > 1, there exist elements S′, S′′ ∈ B

such that S′ ⊔ S′′ = S. Let FS1 , . . . , FSk, k > 3, be a minimal collection of facets

that intersect pairwise but have empty common intersection. We shall lead thisto a contradiction by finding a nontrivial subcollection of FS1 , . . . , FSk

with emptycommon intersection.

Assume there is a set S ∈ B|S intersecting more than one Si, but not intersect-

ing every Si. Then the subcollection of facets FSisatisfying Si ∩ S 6= ∅ will have

empty common intersection, since⊔

Si : Si∩S 6=∅

Si ∈ B

by definition of a building set.

It remains to find S ∈ B|S intersecting more than one Si, but not intersectingevery Si. By Theorem 1.5.13, S1 ⊔ · · · ⊔ Sk = S ∈ B. Therefore, we can writeS = S′⊔S′′, where S′, S′′ ∈ B. Let S1 be that of the sets S′ and S′′ which intersectsmore elements Si than the other. Then S1 intersects more than one Si. If S1 doesnot intersect all of Si, then we are done. Otherwise we write S1 = S′1 ⊔S′′1, whereS′1, S′′1 ∈ B, and choose as S2 that of the sets S′1 and S′′1 which intersects moreelements Si than the other. If S2 intersects all the sets Si, choose S3 in the same


way, and so on. Since the cardinality of the sets S1, S2, S3, . . . strictly decreases,at some point this process stops, and we get that one of the sets S′i, S′′i intersectsmore than one Si, but does not intersect every Si.

Proposition 1.6.14 ([266]). Let B be a building set on [n+1] such that PB isa flagtope. Then there exists a building set B0 ⊂ B such that PB0 is a combinatorialcube with dimPB0 = dimPB.

Proof. By Proposition 1.5.2, we need to consider only connected building sets.For n = 1, the proposition is true. Assuming that the assertion holds for m < n,we shall prove it for m = n. By Proposition 1.6.13, we have [n + 1] = S′ ⊔ S′′,where S′, S′′ ∈ B. By the induction assumption, the building sets B|S′ and B|S′′

have subsets B′0 and B′′

0 whose corresponding nestohedra are cubes. The buildingset B0 = (B′

0 ⊔ B′′0 ) ∪ [n+ 1] is the desired one (see Example 1.5.22).

It now follows from Lemma 1.5.17 that a flag nestohedron can be obtained froma cube by a sequence of face truncations. The following lemma shows that there isa sequence consisting only of codimension-2 face truncations:

Lemma 1.6.15. Let B1 ⊂ B2 be connected building sets on [n+ 1] whose corre-sponding nestohedra PB1 and PB2 are flagtopes. Then

(a) PB2 is obtained from PB1 by a sequence of 2-truncations;(b) γi(PB1) 6 γi(PB2) for i = 0, 1, . . . , [n/2].

Furthermore, if B1 6= B2, then the inequality above is strict for some i.

Proof. Let S be a minimal (by inclusion) element of B2 \ B1. We set B′ =B1 ∪ S to be the minimal (by inclusion) building set containing B1 ∪ S. By

Proposition 1.6.13, there exist S′, S′′ ∈ B2 such that S′ ⊔ S′′ = S. It follows fromthe choice of S that S′, S′′ ∈ B1. It is easy to show that B′ is the collection of setsB1 ∪ T = T ′ ⊔ T ′′ : T ′, T ′′ ∈ B1, S′ ⊂ T ′, S′′ ⊂ T ′′. Hence, the decomposition ofany element of B′ \ B1 consists of two elements. Therefore, by Lemma 1.5.17, thenestohedron PB′ is obtained from PB1 by a sequence of 2-truncations.

Since B1 ( B′ ⊂ B2, we can finish the proof of (a) by induction on the numberof elements in B2 \ B1. Statement (b) follows from Corollary 1.6.10.

Finally we can prove the main result of this section, giving a characterisationof flag nestohedra:

Theorem 1.6.16 ([64]). A nestohedron PB is a flagtope if and only if it is a2-truncated cube.

More precisely, if PB is a flagtope, then there exists a sequence of building setsB0 ⊂ B1 ⊂ · · · ⊂ BN = B, where PB0 is a combinatorial cube, Bi = Bi−1 ∪ Si,and PBi

is obtained from PBi−1 by 2-truncation at the face FSj1∩ FSj2

⊂ PBi−1 ,where Si = Sj1 ⊔ Sj2 , and Sj1 , Sj2 ∈ Bi−1.

Proof. By Proposition 1.5.23, we need to consider only connected buildingsets. The ‘only if’ statement follows from Proposition 1.6.14 and Lemma 1.6.15.To prove the ‘if’ statement we need to check that a polytope obtained from aflagtope by a 2-truncation is a flagtope. This is left as an exercise.

Together with Proposition 1.6.12, Theorem 1.6.16 implies


Corollary 1.6.17. The Gal conjecture holds for all flag nestohedra PB, i.e.γi(PB) > 0.

Example 1.6.18. Let us see how Theorem 1.6.16 works in the case of the3-dimensional associahedron. The building set corresponding to As3 is given by

B =1, 2, 3, 4, 1, 2, 2, 3, 3, 4, 1, 2, 3, 2, 3, 4, 1, 2, 3, 4

(see Fig. 1.5, right). In order to obtain As3 from I3 by 2-truncations, we have tospecify a building set B0 ⊂ B, such that PB0 ≈ I3, and to order the elements ofB \ B0 in such a way that adding a new element to the building set corresponds toa 2-truncation.

First let B0 consist of i, 1, 2, 3, 4, [4]. The associahedron PB is then ob-tained from PB0 ≈ I3 by consecutive truncation at the faces F1,2 ∩ F3, F2 ∩F3,4, F2 ∩ F3 in this order. (Warning: PB0 is not the rectangular paral-lelepiped, it is a tetrahedron with two opposite edges truncated. However the three2-truncations of PB0 described here and the three 2-truncations of the rectangularparallelepiped described in Example 1.5.27 give the same (up to affine equivalence)polytope As3 shown in Fig. 1.5, right.)

Another option is to let B0 consist of i, 1, 2, 1, 2, 3, [4]. Then PB0 isobtained from a tetrahedron by truncating a vertex and then truncating an edgewhich contained this vertex; we have that PB0 ≈ I3. To obtain the associahedronPB from PB0 we first truncate the face F2 ∩ F3 of PB0 ≈ I3 and get the newfacet F2,3. Then we truncate the faces F2,3 ∩ F4 and F3 ∩ F4.

Exercises.

1.6.19. In a flagtope, a collection of pairwise intersecting faces (of any dimen-sion) has a nonempty intersection.

1.6.20. A face of a flagtope is a flagtope. This generalises Proposition 1.6.3.

1.6.21. Give an example of a triangle-free simple n-polytope with 2n+3 facetswhich is not a flagtope.

1.6.22. Give an example of a non-flag simple polytope P with γi(P ) > 0.

1.6.23. A polytope obtained from a flagtope by a 2-truncation is a flagtope.

1.6.24 ([64, Theorem 9.1]). The face vectors of n-dimensional flag nestohedraPB satisfy

(a) γi(In) 6 γi(PB) 6 γi(Pe

n) for i = 0, 1, . . . , [n/2];(b) gi(I

n) 6 gi(PB) 6 gi(Pen) for i = 0, 1, . . . , [n/2];

(c) hi(In) 6 hi(PB) 6 hi(Pe

n) for i = 0, 1, . . . , n;(d) fi(I

n) 6 fi(PB) 6 fi(Pen) for i = 0, 1, . . . , n.

Furthermore, the lower bounds are achieved only for PB = In and the upper boundsare achieved only for PB = Pen. (Hint: to prove (a) use Lemma 1.6.15. The otherinequalities follow from Proposition 1.3.13.)

1.6.25 ([64, Theorem 9.2]). The face vectors of graph associahedra PΓ corre-sponding to connected graphs Γ on [n+ 1] satisfy

(a) γi(Asn) 6 γi(PΓ) 6 γi(Pe

n) for i = 0, 1, . . . , [n/2];(b) gi(As

n) 6 gi(PΓ) 6 gi(Pen) for i = 0, 1, . . . , [n/2];

(c) hi(Asn) 6 hi(PΓ) 6 hi(Pe

n) for i = 0, 1, . . . , n;


(d) fi(Asn) 6 fi(PΓ) 6 fi(Pe

n) for i = 0, 1, . . . , n.

Furthermore, the lower bounds are achieved only for PB = Asn and the upperbounds are achieved only for PB = Pen.

1.7. Differential algebra of combinatorial polytopes

Here we develop a differential-algebraic formalism that will allow us to analysethe combinatorics of families of polytopes and their face numbers from the viewpointof differential equations. All polytopes in this section are combinatorial.

Ring of polytopes. Denote by P2n the free abelian group generated by allcombinatorial n-polytopes. The group P2n is infinitely generated for n > 1, and itsplits into a direct sum of finitely generated components as follows:

P2n =⊕

m>n+1

P2n,2(m−n)

where P2n,2(m−n) is the group generated by n-polytopes with m facets. (The factthat there are finitely many combinatorial polytopes with a given number of facetsis left as an exercise.)

Definition 1.7.1. The product of polytopes turns the direct sum

P =⊕

n>0

P2n = P0 +⊕

m> 2

m−1⊕

n=1

P2n,2(m−n)

into a bigraded commutative ring, the ring of polytopes. The unit is P 0, a point.Simple polytopes generate a bigraded subring of P, which we denote by S.

A polytope is indecomposable if it cannot be represented as a product of twoother polytopes of positive dimension.

Proposition 1.7.2 ([50, Proposition 2.22]). P is a polynomial ring generatedby indecomposable combinatorial polytopes.

Proof. We need to show that any polytope P of positive dimension can berepresented as a product of indecomposable polytopes, P = P1 × · · · × Pk, andthis representation is unique up to reordering the factors. The existence of such adecomposition is clear. Now assume that P1×· · ·×Pk ≈ Q1×· · ·×Qs. Fix a vertexv = v1×· · ·× vk = w1×· · ·×ws, where vi ∈ P and wj ∈ Qj . The faces of the formv1 × · · · × Pi × · · · × vk and w1 × · · · ×Qj × · · · × ws are maximal indecomposablefaces of P containing the vertex v, and therefore they bijectively correspond to eachother under the combinatorial equivalence. It follows that k = s and Pi ≈ Qi fori = 1, . . . , k up to a permutation of factors.

Since P is a free abelian group generated by combinatorial polytopes, and eachpolytope can be uniquely represented by a monomial in indecomposable polytopes,it follows that P is a polynomial ring on indecomposable polytopes.

Given P ∈ P2n, denote by dP ∈ P2(n−1) the sum of all facets of P in thering P. The following lemma is straightforward.

Lemma 1.7.3. d : P→ P is a linear operator of degree −2 satisfying the identity

d(P1P2) = (dP1)P2 + P1(dP2).

Therefore, P is a differential ring, and S is its differential subring.

1.7. DIFFERENTIAL ALGEBRA OF COMBINATORIAL POLYTOPES 51

Example 1.7.4. We have

dIn = n(dI)In−1 = 2nIn−1, and d∆n = (n+ 1)∆n−1.

Face-polynomials revisited. By Proposition 1.3.6, the F -polynomial andthe H-polynomial define ring homomorphisms

F : P −→ Z[s, t], H : P −→ Z[s, t],

which send P ∈ P to F (P )(s, t) and H(P )(s, t) respectively.

Theorem 1.7.5. For any simple polytope P we have

(1.29) F (dP ) =∂

∂tF (P ).

Proof. Assume that P is a simple n-polytopse with facets P1, . . . , Pm. Then

F (dP ) =m∑

i=1

F (Pi) =m∑

i=1

n−1∑

k=0

fk(Pi)sktn−1−k.

On the other hand,

∂

∂tF (P ) =

n−1∑

k=0

(n− k)fk(P )sktn−1−k.

Comparing the coefficients in the two sums above we reduce (1.29) to

(1.30)

m∑

i=1

fk(Pi) = (n− k)fk(P ).

Since P is simple, every k-face is contained in exactly n− k facets, so it is countedn− k times on the left hand side of the above identity.

Corollary 1.7.6. Let P1 and P2 be two simple n-polytopes such that dP1 =dP2 in S. Then F (P1) = F (P2).

Proof. Indeed, F (dP1) = F (dP2) implies that ∂F (P1)∂t = ∂F (P2)

∂t . Using thatF (P )(s, 0) = sn we obtain F (P1) = F (P2).

Proposition 1.7.7. Let F : S→ Z[s, t] be a linear map such that

F (dP ) =∂

∂tF (P ) and F (P )|t=0 = sn.

Then F (P ) = F (P ).

Proof. We have F (P 0) = 1 = F (P 0). Assume by induction the state-ment is true in dimensions 6 n − 1, and let P be a simple n-polytope. Then

F (dP )(s, t) = F (dP )(s, t). It follows that ∂∂t F (P ) = ∂

∂tF (P ). Therefore,

F (P )(s, t) = F (P )(s, t) + C(s). Setting t = 0, we obtain sn = sn + C(s), whenceC(s) = 0.

Theorem 1.7.5 also allows us to reduce the Dehn–Sommerville relations (1.12)to the Euler formula:

Theorem 1.7.8. The following identity holds for any simple n-polytope P :

(1.31) F (P )(s, t) = F (P )(−s, s+ t).


Proof. We have F (P 0)(s, t) = 1 = F (P 0)(−s, s + t). Assume by inductionthe identity holds in dimensions 6 n − 1. Then for a given P of dimension n wehave F (dP )(s, t) = F (dP )(−s, s+ t). Therefore, ∂

∂tF (P )(s, t) =∂∂tF (P )(−s, s+ t)

and F (P )(s, t)− F (P )(−s, s+ t) = C(s). By Euler’s formula (1.11),

F (−s, s) =(f0 − f1 + · · ·+ (−1)nfn

)sn = sn.

Therefore, C(s) = F (P )(s, 0)− F (P )(−s, s) = 0.

Example 1.7.9. For a simple 3-polytope P 3 with m = f2 facets we havef1 = 3(m− 2), f0 = 2(m− 2). Assume that all facets are k-gons, so dP 3 = mP 2

k .Then by Theorem 1.7.5,

m(s2 + kst+ kt2) =∂

∂t

(s3 +ms2t+ 3(m− 2)st2 + 2(m− 2)t3

),

which implies m(6 − k) = 12. It follows that the pair (k,m) may only take values(3, 4), (4, 6) and (5,12) corresponding to a simplex, cube and dodecahedron.

For general polytopes the difference δ = Fd − ∂∂tF measures the failure of

identity (1.29).A polytope is said to be k-simple (for k > 0) if each of its k-faces is an inter-

section of exactly n − k facets. For example, a simple polytope is 0-simple, while1-simple polytopes are also known as simple in edges. Every n-polytope is (n− 1)-and (n− 2)-simple.

A polytope is k-simplicial if each of its k-dimensional faces is a simplex. Asimplicial n-polytope is (n − 1)-simplicial, and every polytope is 1-simplicial. Bypolarity, if an n-polytope P is k-simple, then P ∗ is (n− 1− k)-simplicial.

Theorem 1.7.10. Let P ∈ P. Then the following identity holds:

F (dP ) =∂

∂tF (P ) + δ(P ),

where δ : P→ Z[s, t] is an F -derivation, i.e. it is linear and satisfies the identity

δ(P1P2) = δ(P1)F (P2) + F (P1)δ(P2).

Moreover, if P is an n-polytope, then δ(P ) = δ2sn−3t2 + · · ·+ δn−1t

n−1 and δi > 0for 2 6 i 6 n− 1. Also, P is k-simple if and only if δn−1−k = 0; in this case δi = 0for 2 6 i 6 n− 1− k.

Proof. The fact that δ = Fd− ∂∂tF is an F -derivation is verified by a direct

computation. By (1.30) the coefficient of sktn−1−k in δ(P ) is given by

δn−1−k =

m∑

i=1

fk(Pi)− (n− k)fk(P ).

It vanishes for k = n − 1 and k = n − 2 (as each codimension-two face of P iscontained in exactly two facets). Also, δn−1−k is non-negative because every k-faceis contained in at least n − k facets. Finally, if P is k-simple, then every j-face iscontained in exactly n− j facets for j > k, so δn−1−j vanishes for j > k.

Example 1.7.11. Let P be a simple n-polytope, and P ∗ the dual simplicialpolytope. Since the face poset of P ∗ is the opposite to the face poset of P ,

F (P ∗)(s, t) = sn + tF (P )(t, s)− tn

s= sn +

n−1∑

k=0

fk(P )sn−1−ktk+1 .

1.7. DIFFERENTIAL ALGEBRA OF COMBINATORIAL POLYTOPES 53

We have dP ∗ = f0(P )∆n−1, therefore,

δ(P ∗) = f0(P )(s+ t)n − tn

s− ∂

∂t

n∑

k=0

fk(P )sn−1−ktk+1.

The coefficient of sn−1−ktk on the right hand side above is given by

δk(P∗) =

(n

k

)f0(P )− (k + 1)fk(P ), for 1 6 k 6 n.

This is non-negative by Theorem 1.7.10 or by Exercise 1.4.15.

The following properties of H(P ) follow from the corresponding properties ofF (P ) established above and the identity H(P )(s, t) = F (P )(s− t, t).

Theorem 1.7.12.

(a) The ring homomorphism H : P −→ Z[s, t] satisfies H(P )∣∣t=0

= sn. Therestriction of H to the subring S of simple polytopes satisfies the equation

H(dP ) = ∂H(P )

where ∂ = ∂∂s +

∂∂t .

(b) The image of H is generated by H(∆1) = s+ t and H(∆2) = s2 + st+ t2.

(c) If H : S −→ Z[s, t] is a linear map satisfying H(dP ) = ∂H(P ) and

H(P )∣∣t=0

= sn for any simple n-polytope P , then H(P ) = H(P ).

Ring of building sets. Let B1 and B2 be building sets on [n1] and [n2] re-spectively. A map f : B1 → B2 of building sets is a map f : [n1] → [n2] satisfyingf−1(S) ∈ B1 for every S ∈ B2. Two building sets B1 and B2 are said to be equiva-lent, if there exist maps f : B1 → B2 and g : B2 → B1 of building sets such that thecompositions f g and g f are the identity maps.

The product of B1 and B2 is the building set B1 · B2 on [n1 + n2] induced byappending the interval [n2] to the interval [n1].

Example 1.7.13. The collection S of Example 1.5.3 and the complete collec-tion C are both connected building sets on [n + 1]. They are initial and terminalconnected building sets respectively, since for every connected building set B on[n + 1] there are maps C → B → S of building sets induced by the identity mapon [n+ 1].

Definition 1.7.14. Denote by B2n the free abelian group generated by theequivalence classes of building sets on [k] for k 6 n+ 1. Since B1 · B2 is equivalentto B2 · B1, the product turns B =

⊕n>0 B

2n into a commutative associative ring.

Given a connected building set B on [n+ 1], define

(1.32) dB =∑

S∈B\[n+1]

B|S · B/S

where the sum is taken in the ring B. Since B is generated by connected buildingsets, we may extend d to a linear map d : B→ B using the Leibniz formula

(1.33) d(B1 · B2) = (dB1) · B2 + B1 · (dB2).We therefore get a derivation of B.


Proposition 1.7.15. The map B 7→ PB induces a differential ring homomor-phism β : B → P. Its image is a graded subring with unit in P multiplicativelygenerated by nestohedra PB corresponding to connected building sets B.

Proof. The fact that β is a ring homomorphism is obvious. It follows fromCorollary 1.5.12 and (1.33) that β commutes with the differentials. Every buildingset consisting only of singletons is mapped to P 0 which represents a unit in P.Finally, it follows from Proposition 1.5.2 (a) that the image β(B) is generated byPB with connected B.

Remark. The ring B does not have unit, and the map β is not injective.

Given a subset S ⊂ [n + 1], denote by Γ|S the restriction of Γ to the vertexset S, and denote by Γ/S the graph with the vertex set [n + 1]\S having an edgebetween two vertices i and j whenever they are path connected in ΓS∪i,j.

Proposition 1.7.16. For a connected graph Γ on n+ 1 vertices, we have

dPΓ =∑

S([n+1]

Γ|S is connected

P (Γ|S)× P (Γ/S).

Proof. Follows directly from (1.32).

Example 1.7.17. We have the following formulae for the differentials of thefour graph-associahedra defined in Section 1.5:

dAsn =∑

i+j=n−1

(i + 2)Asi ×Asj ;

dPen =∑

i+j=n−1

(n+ 1

i+ 1

)Pei × Pej ;

dCyn = (n+ 1)∑

i+j=n−1

As i × Cyj ;

dStn = n · Stn−1 +

n−1∑

i=0

(n

i

)Sti × Pen−i−1.

For example (see Figs. 1.4–1.8),

dAs3 = 2As0 ×As2 + 3As1 ×As1 + 4As2 ×As0 ;

dPe3 = 4Pe0 × Pe2 + 6Pe1 × Pe1 + 4Pe2 × Pe0 ;

dCy3 = 4(As0 × Cy2 +As1 × Cy1 +As2 × Cy0) ;dSt3 = 3St2 + St0 × Pe2 + 3St1 × Pe1 + 3St2 × Pe0 .

Exercises.

1.7.18. Show that there are finitely many combinatorial polytopes with a givennumber of facets.

1.7.19. Show that δ = Fd− ∂∂tF is an F -derivation.

1.8. FAMILIES OF POLYTOPES AND DIFFERENTIAL EQUATIONS 55

1.8. Families of polytopes and differential equations

In this section, using the language of generating series, we interpret the for-mulae for the differential of nestohedra and graph-associahedra as certain partialdifferential equations. These differential equations encode the combinatorial infor-mation of the face structure of nestohedra. This exposition builds upon the resultsof [51] and [48].

Denote by P[q] the polynomial ring in an indeterminate q with coefficients in P.

Proposition 1.8.1. Let

Q : P→ P[q], P 7→ Q(P ; q)

be a linear map such that

Q(dP ; q) =∂

∂qQ(P ; q) and Q(P ; 0) = P

for any polytope P . Then

Q(P ; q) =

n∑

k=0

(dkP )qk

k!.

Proof. Use induction by dimension, as in the proof of Proposition 1.7.7.

Now assume given a sequence P = Pn, n > 0 of polytopes, one in eachdimension. We define its generating series as the formal power series

P(x) =∑

n>0

λnPnxn+n0

in P ⊗ Q[[x]]. The parameter n0 and the coefficients λn will be chosen dependingon a particular sequence P . Using the transformation Q of the previous propositionwe may define the following 2-parameter extension of the generating series:

(1.34) P(q, x) =∑

n>0

λnQ(Pn; q)xn+n0 .

We have P(0, x) = P(x).We consider the following generating series of the six sequences of nestohedra:

(1.35)

∆(x) =∑

n>0

∆n xn+1

(n+ 1)!; I(x) =

∑

n>0

Inxn

n!;

As(x) =∑

n>0

Asnxn+2 ; Pe(x) =∑

n>0

Penxn+1

(n+ 1)!;

Cy(x) =∑

n>0

Cynxn+1

n+ 1; St(x) =

∑

n>0

Stnxn

n!.

Lemma 1.8.2. The differentials of the generating series above are given by

d∆(x) = x∆(x) ; dI(x) = 2xI(x) ;

dAs(x) = As(x)d

dxAs(x) ; dPe(x) = Pe2(x) ;

dCy(x) = As(x)d

dxCy(x) ; dSt(x) =

(x+ Pe(x)

)St(x) .


Proof. This follows from the formulae of Examples 1.7.4 and 1.7.17.

Theorem 1.8.3. The two-parameter extensions of the generating series (1.35)satisfy the following partial differential equations:

∂

∂q∆(q, x) = x∆(q, x) ;

∂

∂qI(q, x) = 2xI(q, x) ;

∂

∂qAs(q, x) = As(q, x)

∂

∂xAs(q, x) ;

∂

∂qPe(q, x) = Pe2(q, x) ;

∂

∂qCy(q, x) = As(q, x)

∂

∂xCy(q, x) ;

∂

∂qSt(q, x) =

(x+ Pe(q, x)

)St(q, x) .

Proof. A direct calculation using formulae (1.34) and (1.35).

Remark. The role of the parameters λn in (1.34) can be illustrated as follows.If we replace the first series ∆(x) of (1.35) by

∆(x) =∑

n>0

∆n xn+1

n+ 1,

then the first equations of Lemma 1.8.2 and Theorem 1.8.3 take the form

d∆(x) = x2d

dx∆(x),

∂

∂q∆(q, x) = x2

d

dx∆(q, x).

Four of the equations of Theorem 1.8.3, namely those corresponding to theseries ∆, I, Pe and St, are ordinary differential equations. Their solutions arecompletely determined by the initial data P(0, x) = P(x) and are given by theexplicit formulae

∆(q, x) = ∆(x)eqx ; I(q, x) = I(x)e2qx ;

Pe(q, x) =Pe(x)

1− qPe(x) ; St(q, x) = St(x)eqx

1− qPe(x) .

The equation for U = As(q, x) has the form Uq = UUx. This classical quasilin-ear partial differential equation was considered by E. Hopf, and therefore becameknown as the Hopf equation.

Theorem 1.8.4.

(a) The series As(q, x) is given by the solution of the functional equation( equation on characteristics)

(1.36) As(q, x) = As(x + qAs(q, x)),

where As(x) = As(0, x).(b) The series Cy(q, x) is given by the solution of

(1.37) Cy(q, x) = Cy(x + qAs(q, x)),

where Cy(x) = Cy(0, x).

Proof. Set U = As(q, x) and Asx = ddxAs(x). If U is a solution to (1.36),

then we obtain by differentiating

Uq = (U + qUq)Asx, Ux = (1 + qUx)Asx.


Therefore, (1 − qAsx)Uq = UAsx and (1 − qAsx)Ux = Asx, which implies thatU satisfies the Hopf equation Uq = UUx. Its solution with the initial conditionU(0, x) = As(x) is unique by the general theory of quasilinear equations (in our casethe uniqueness can be also verified using standard arguments with power series).

Similarly, by differentiating (1.37) we obtain for V = Cy(q, x):

Vq = (U + qUq)Cyx, Vx = (1 + qUx)Cyx.

Using that Uq = UUx we rewrite the first equation above as Vq = U(1 + qUx)Cyx,which implies Vq = UVx as claimed. This is exactly the equation for V = Cy(q, x)given by Theorem 1.8.3, and its solution with V (0, x) = Cy(x) is unique.

We can also use Lemma 1.8.2 to calculate the face-polynomials F (s, t) of graph-associahedra. Let P(x) be one of the generating series (1.35), and set

FP = F (P(x)) =∑

n>0

λnxn+n0

n∑

k=0

fk(Pn)sktn−k.

We refer to FP = FP (s, t;x) as the generating series of face-polynomials; it is aseries in x whose coefficients are polynomials in s and t.

Theorem 1.8.5. The generating series of face-polynomials correspondingto (1.35) satisfy the following differential equations, with the initial conditionsgiven in the second column:

∂

∂tF∆ = xF∆, F∆(s, 0;x) =

esx − 1

s;

∂

∂tFI = 2xFI , FI(s, 0;x) = esx ;

∂

∂tFAs = FAs

∂

∂xFAs, FAs(s, 0;x) =

x2

1− sx ;

∂

∂tFPe = F 2

Pe, FPe(s, 0;x) =esx − 1

s;

∂

∂tFCy = FAs

∂

∂xFCy, FCy(s, 0;x) = −

ln(1− sx)s

;

∂

∂tFSt =

(x+ FPe

)FSt, FSt(s, 0;x) = esx .

Proof. The differential equations are obtained by applying F to the equationsof Lemma 1.8.2 and using the fact that F (dP ) = ∂

∂tF (P ). The initial conditionsfollow by substituting sn for Pn in (1.35) and calculating the resulting series.

Again, the four equations for the series F∆, FI , FPe and FSt are ordinarydifferential equations. Their solutions are completely determined by the initial data;the explicit formulae are left as exercises. The remaining two partial differentialequations for the series FAs and FCy can be explicitly solved as follows.

Theorem 1.8.6.

(a) The series U = FAs satisfies the quadratic equation

(1.38) t(s+ t)U2 + (2tx+ sx− 1)U + x2 = 0.

The initial condition FAs(s, 0;x) =x2

1−sx determines its solution uniquely.


(b) The series FCy is given by

FCy = −1

sln(1− s(x+ tFAs ))

).

Proof. (a) By analogy with (1.36) we show that U = FAs satisfies

U = ϕ(x+ tU),

where ϕ(x) = FAs(s, 0;x) =x2

1−sx . It is equivalent to (1.38).

(b) We have that V = FCy is given by the solution to Vt = UVx. By analogywith (1.37) we show that it is given by

V = ψ(x+ tU),

where U = FAs and ψ(x) = FCy(s, 0;x) = − ln(1−sx)s .

As a corollary, we shall derive a formula for the number of k-faces in Asn,which equals the number of bracketing of a word of n+ 2 letters with n− k pairsof brackets. These numbers were first calculated by Cayley in 1891:

Theorem 1.8.7. The number of k-dimensional faces in an n-dimensional as-sociahedron is given by

fk(Asn) =

1

n+ 2

(n

k

)(2n− k + 2

n+ 1

).

Proof. We use the fact that FAs satisfies the Hopf equation, whose solutionsmay be obtained using conservation laws. Let

U(t, x) =∑

k>0

Uk(x)tk

be the solution of the Cauchy problem for the Hopf equation:

(1.39) Ut = UUx, U(0, x) = ϕ(x).

This equation has the following conservation laws(Uk

k

)

t

=

(Uk+1

k + 1

)

x

, for k > 1.

Hence,

dk

dtkU =

dk−1

dtk−1

(U2

2

)

x

=dk−2

dtk−2

(U3

3

)

xx

= · · · = dk

dxk

(Uk+1

k + 1

), for k > 1.

Evaluating at t = 0 we obtain

dk

dtkU

∣∣∣∣t=0

= k!Uk(x) =dk

dxk

(Uk+10 (x)

k + 1

).

Therefore,

Uk(x) =1

(k + 1)!

dk

dxkϕk+1(x).

By Theorem 1.8.5 the function

(1.40) U = FAs(s, t;x) =∑

n>0

n∑

k=0

fn−k(Asn)sn−ktkxn+2


satisfies the Hopf equation (1.39) with the initial function ϕ(s;x) = x2

1−sx . Wetherefore calculate

Uk(s;x) =1

(k + 1)!

dk

dxk

(x2(k+1)

(1 − sx)k+1

)

=1

(k + 1)!

dk

dxk

(x2(k+1)

∑

l>0

(l + k

l

)slxl

)

=1

(k + 1)!

∑

l>0

(l + k

k

)(2k + l+ 2)!

(k + l + 2)!slxk+l+2

=∑

n>k

1

n+ 2

(n

k

)(n+ k + 2

n+ 1

)sn−kxn+2.

On the other hand, it follows from (1.40) that

Uk(s;x) =∑

n>k

fn−k(Asn)sn−kxn+2.

Comparing the last two formulae we obtain

fn−k(Asn) =

1

n+ 2

(n

k

)(n+ k + 2

n+ 1

),

which is equivalent to the required formula.

Exercises.

1.8.8. By solving the first two differential equations of Theorem 1.8.5 show thatthe generating series for the F -polynomials of simplices and cubes are given by

F∆(s, t;x) =∑

n>0

F (∆n)xn+1

(n+ 1)!= etx

esx − 1

s,

FI(s, t;x) =∑

n>0

F (In)xn

n!= etxe(s+t)x.

Compare this with the formulae for F (∆n) and F (In).

1.8.9. Show by solving the corresponding differential equations from Theo-rem 1.8.5 that the generating series for the face-polynomials of permutahedra andstellahedra are given by

FPe(s, t;x) =∑

n>0

F (Pen)xn+1

(n+ 1)!=

esx − 1

s− t(esx − 1),

FSt (s, t;x) =∑

n>0

F (Stn)xn

n!=

se(s+t)x

s− t(esx − 1).

Compute the face-polynomials F (Pen) and F (Stn) and the face numbers explicitly.

1.8.10. Define the generating series HP(s, t;x) of H-polynomials by analogywith the generating series of face-polynomials. Deduce the following formulae for


sequences of nestohedra:

H∆(s, t;x) =∑

n>0

H(∆n)xn+1

(n+ 1)!=esx − etxs− t ,

HI(s, t;x) =∑

n>0

H(In)xn

n!= e(s+t)x,

HPe(s, t;x) =∑

n>0

H(Pen)xn+1

(n+ 1)!=

esx − etxsetx − tesx ,

HSt (s, t;x) =∑

n>0

H(Stn)xn

n!=

(s− t)e(s+t)xsetx − tesx .

1.8.11. The series Y = HAs (s, t;x) =∑

n>0H(Asn)xn+2 satisfies the quadraticequation

Y = (x+ sY )(x + tY ).

The initial condition HAs (s, 0;x) =x2

1−sx determines its solution uniquely.

1.8.12. The components of the h-vector of Asn are given by

hk =1

n+ 1

(n+ 1

k

)(n+ 1

k + 1

), 0 6 k 6 n.

CHAPTER 2

Combinatorial structures

The face poset of a convex polytope is a classical example of a combinatorialstructure underlying a decomposition of a geometric object. With the developmentof combinatorial topology several new combinatorial structures emerged, such assimplicial and cubical complexes, simplicial posets and other types of regular cellu-lar (or CW) complexes. Many of these structures have eventually become objectsof independent study in geometric combinatorics.

A simplicial complex is the abstract combinatorial structure behind a simpli-cial subdivision (or triangulation) of a topological space. Triangulations were firstintroduced by Poincare and provide a rigorous and convenient tool for studyingtopological invariants of smooth manifolds by combinatorial methods. The notionof a nerve of a covering of a topological space, introduced by Alexandroff, providesanother source of examples of simplicial complexes.

The study of triangulations stimulated the development of first combinatorialand then algebraic topology in the first half of the XXth century. With the ap-pearance of cellular complexes algebraic tools gradually replaced the combinatorialones in mainstream topology. Simplicial complexes still play a pivotal role in PL(piecewise linear) topology, however nowadays the main source of interest in themis in discrete and computational geometry. One reason for that is the emergence ofcomputers, since simplicial complexes provide the most effective way to translategeometric and topological structures into machine language.

We therefore may distinguish two different views on the role of simplicial com-plexes and triangulations. In topology, simplicial complexes and their differentderivatives such as singular chains and simplicial sets are used as technical toolsto study the topology of the underlying space. Most combinatorial invariants ofnerves or triangulations (such as the number of faces of a given dimension) do nothave meaning in topology, as they do not reflect any topological feature of theunderlying space. Topologists therefore tend not to distinguish between simplicialcomplexes that have the same underlying topology. For instance, refining a trian-gulation (such as passing to the barycentric subdivision) changes the combinatoricsdrastically, but does not affect the underlying topology. On the other hand, incombinatorial geometry the combinatorics of a simplicial complex is what reallymatters, while the underlying topology is often simple or irrelevant.

In toric topology the combinatorist’s point of view on triangulations and simi-lar decompositions is enriched by elaborate topological techniques. Combinatorialinvariants of triangulations therefore can be analysed by topological methods, andat the same time combinatorial structures such as simplicial complexes or posetsbecome a source of examples of topological spaces and manifolds with nice featuresand lots of symmetry, e.g. bearing a torus action. The combinatorial structuresare the subject of this chapter; the associated topological objects will come later.

61

62 2. COMBINATORIAL STRUCTURES

Here we assume only minimal knowledge of topology. The reader may alsocheck Appendix B for the definition of simplicial homology groups, etc.

2.1. Polyhedral fans

Like convex polytopes, polyhedral fans encode both geometrical and combina-torial information. The geometry of fans is poorer than that of convex polytopes,but this geometry is still a part of the structure of a fan, which distinguishes fansfrom the purely combinatorial objects considered later in this chapter.

Although fans were considered in convex geometry independently, the mainsource of interest to them is in the theory of toric varieties, which are classified byrational fans. Toric varieties are the subject of Chapter 5, and here we describe theterminology and constructions related to fans.

Definition 2.1.1. A set of vectors a1, . . . ,ak ∈ Rn defines a convex polyhedralcone, or simply cone,

σ = R>〈a1, . . . ,ak〉 = µ1a1 + · · ·+ µkak : µi ∈ R>.Here a1, . . .ak are referred to as generating vectors (or generators) of σ. A minimalset of generators of a cone is defined up to multiplication of vectors by positiveconstants. A cone is rational if its generators can be chosen from the integer latticeZn ⊂ Rn. If σ is a rational cone, then its generators a1, . . .ak are usually chosento be primitive, i.e. each a i is the smallest lattice vector in the ray defined by it.

A cone is strongly convex if it does not contain a line. A cone is simplicial if itis generated by a part of basis of Rn, and is regular if it is generated by a part ofbasis of Zn. (Regular cones play a special role in the theory of toric varieties, seeChapter 5.)

A cone is also an (unbounded) intersection of finitely many halfspaces in Rn, soit is a convex polyhedron in the sense of Definition 1.1.2. We therefore may defineface of a cone in the same way as we did for polytopes, as intersections of σ withsupporting hyperplanes. We can only consider supporting hyperplanes containing 0;such a hyperplane is defined by a linear function u and will be denoted by u⊥. Aface τ of a cone σ is therefore an intersection of σ with a supporting hyperplaneu⊥, i.e. τ = σ∩u⊥. Every face of a cone is itself a cone. If σ 6= Rn, then σ has thesmallest face σ ∩ (−σ); it is a vertex 0 if and only if σ strongly convex. A minimalgenerator set of a cone consists of nonzero vectors along its edges.

A fan is a finite collection Σ = σ1, . . . , σs of strongly convex cones in someRn such that every face of a cone in Σ belongs to Σ and the intersection of any twocones in Σ is a face of each. A fan Σ is rational (respectively, simplicial, regular) ifevery cone in Σ is rational (respectively, simplicial, regular). A fan Σ = σ1, . . . , σsis called complete if σ1 ∪ · · · ∪ σs = Rn.

Given a cone σ ⊂ Rn, define its dual as

(2.1) σv =x ∈ Rn : 〈u , x〉 > 0 for all u ∈ σ.

(Note the difference with the definition of the polar set of a polyhedron, see (1.4).)Observe that if u ∈ σv , then u⊥ is a supporting hyperplane of σ. It can be shownthat σv is indeed a cone, (σv)v = σ, and σv is strongly convex if and only ifdimσ = n.

Cones in a fan can be separated by hyperplanes (or linear functions):

2.1. POLYHEDRAL FANS 63

Lemma 2.1.2 (Separation Lemma). Let σ and σ′ be two cones whose intersec-tion τ is a face of each. Then there exists u ∈ σv ∩ (−σ′)v such that

τ = σ ∩ u⊥ = σ′ ∩ u⊥.

In other words, a supporting hyperplane defining τ can be chosen so as to separateσ and σ′.

Proof. We only sketch a proof; the details can be found, e.g. in [122, §1.2].The fact that σ and σ′ intersect in a face implies that the cone ξ = σ−σ′ = σ+(−σ′)is not the whole space. Let u⊥ be a supporting hyperplane defining the smallestface of ξ, that is,

ξ ∩ u⊥ = ξ ∩ (−ξ) = (σ − σ′) ∩ (σ′ − σ).We claim that this u has the required properties. Indeed, σ ⊂ ξ implies u ∈ σv ,and τ ⊂ ξ∩ (−ξ) implies τ ⊂ σ∩u⊥. Conversely, if x ∈ σ∩u⊥, then x is in σ′−σ,so that x = y ′− y for y ′ ∈ σ′, y ∈ σ. Then x + y ∈ σ ∩σ′ = τ , which implies thatboth x and y are in τ . Hence σ ∩ u⊥ = τ , and the same argument for −u showsthat u ∈ (−σ′)v and σ′ ∩ u⊥ = τ .

Miraculously, the convex-geometrical separation property above will translateinto topological separation (Hausdorffness) of algebraic varieties and topologicalspaces constructed from fans in the latter chapters.

The next construction assigns a complete fan to every convex polytope.

Construction 2.1.3 (Normal fan). Let P be a polytope (1.1) with m facetsF1, . . . , Fm and normal vectors a1, . . . ,am. Given a face Q ⊂ P define the cone

(2.2) σQ = u ∈ Rn : 〈u , x ′〉 6 〈u , x 〉 for all x ′ ∈ Q and x ∈ P.The dual cone σv

Q is generated by vectors x − x ′ where x ∈ P and x ′ ∈ Q. In otherwords, σv

Q is the “polyhedral angle” at the face Q consisting of all vectors pointingfrom points of Q to points of P .

We say that a vector a i is normal to the face Q if Q ⊂ Fi. The cone σQ isgenerated by those a i which are normal to Q (this is an exercise). Then

ΣP = σQ : Q is a face of Pis a complete fan ΣP in Rn (this is another exercise), which is denoted by ΣP andis referred to as the normal fan of the polytope P . If 0 is contained in the interiorof P then ΣP may be also described as the set of cones over the faces of the polarpolytope P ∗ (yet another exercise).

It is clear from the above descriptions that the normal fan ΣP is simplicial ifand only if P is simple. In this case the definition of ΣP may be simplified: thecones of ΣP are generated by those sets a i1 , . . . ,a ik for which the intersectionFi1 ∩ · · · ∩ Fik is nonempty.

If all vertices of P are in the lattice Zn then the normal fan ΣP is rational,but the converse is not true. Polytopes whose normal fans are regular are calledDelzant (this name comes from a symplectic geometry construction discussed inSection 5.5). Therefore, P is a Delzant polytope if and only if for every vertexx ∈ P the normal vectors to the facets meeting at x can be chosen to form a basisof Zn. In this definition one may replace ‘normal vectors to the facets meeting at x ’by ‘vectors along the edges meeting at x ’. A Delzant polytope is necessarily simple.


The normal fan ΣP of a polytope P contains the information about the normalsto the facets (the generators a i of the edges of ΣP ) and the combinatorial structureof P (which sets of vectors ai span a cone of ΣP is determined by which facetsintersect at a face), however the scalars bi in (1.1) are lost. Not any complete fancan be obtained by ‘forgetting the numbers bi’ from a presentation of a polytope byinequalities, i.e. not any complete fan is a normal fan. This is fails even for regularfans, as is shown by the next example, which is taken from [122].

Example 2.1.4. Consider the complete three-dimensional fan Σ with 7 edgesgenerated by the vectors a1 = e1, a2 = e2, a3 = e3, a4 = −e1 − e2 − e3,a5 = −e1 − e2, a6 = −e2 − e3, a7 = −e1 − e3, and 10 three-dimensional coneswith vertex 0 over the faces of the triangulated boundary of the tetrahedron shownin Fig. 2.1. It is easy to verify that Σ is regular.

12

3

4

5

67

Figure 2.1. Complete regular fan not coming from a polytope.

Assume that Σ = ΣP is the normal fan of a polytope P . Consider the functionψ : Rn → R given by

ψ(u) = minx∈P〈u , x 〉 = min

v∈V (P )〈u , v 〉,

where V (P ) is the set of vertices of P . This function is continuous and its re-striction to every 3-dimensional cone of ΣP is linear. Indeed, 3-dimensional conesσv correspond to vertices v ∈ V (P ), and we have ψ(u) = 〈u , v〉 for u ∈ σv bydefinition (2.2) of σv .

Now consider the two 3-dimensional cones of ΣP generated by the triplesa1,a3,a5 and a1,a5,a6, and let v and v ′ be the corresponding vertices of P .Then ψ(a1) = 〈a1, v 〉, ψ(a3) = 〈a3, v 〉, ψ(a5) = 〈a5, v〉, ψ(a6) = 〈a6, v

′〉, hence,

ψ(a1) + ψ(a5)− ψ(a3) = 〈a1 + a5 − a3, v〉 = 〈a6, v〉 > 〈a6, v′〉 = ψ(a6).

Therefore,ψ(a1) + ψ(a5) > ψ(a3) + ψ(a6).

Similarly,

ψ(a2) + ψ(a6) > ψ(a1) + ψ(a7),

ψ(a3) + ψ(a7) > ψ(a2) + ψ(a5).

Adding the last three inequalities together we get a contradiction.

2.2. SIMPLICIAL COMPLEXES 65

Exercises.

2.1.5. Let σ be a cone in Rn. Show that σv is also a cone, (σv)v = σ, and σv

is strongly convex if and only if dimσ = n.

2.1.6. The cone σQ given by (2.2) is generated by those vectors amonga1, . . . ,am which are normal to Q.

2.1.7. The set σQ : Q is a face of P is a complete fan in Rn.

2.1.8. If 0 is contained in the interior of P then ΣP consists of cones over thefaces of the polar polytope P ∗.

2.1.9. Let P be a convex polytope (not necessarily simple). Does the collectionof cones generated by the sets a i1 , . . . ,a ik of normal vectors for which Fi1 ∩· · ·∩Fik 6= ∅ form a fan?

2.2. Simplicial complexes

A simplex is the convex hull of a set of affinely independent points in Rn.

Definition 2.2.1. A geometric simplicial complex in Rn is a collection P ofsimplices of arbitrary dimension such that every face of a simplex in P belongs toP and the intersection of any two simplices in P is either empty or a face of each.

To make the exposition more streamlined and without creating much ambigu-ity we shall not distinguish between the collection P (which is an abstract set ofsimplices) and the union of its simplices (which is a subset in Rn). In PL (piece-wise linear) topology the latter union is usually referred to as ‘the polyhedron of P ’.Although we have already reserved the term ‘polyhedron’ for a finite intersection ofhalfspaces (1.1), we shall also occasionally use it in the PL topological sense (whenit creates no ambiguity).

A face of P is a face of any of its simplices. The dimension of P is the maximaldimension of its faces.

If we know the set of vertices of P in Rn then we may recover the whole P byspecifying which subsets of vertices span simplices. This observation leads to thefollowing definition.

Definition 2.2.2. An abstract simplicial complex on a set V is a collectionK of subsets I ⊂ V such that if I ∈ K then any subset of I also belongs to K.We always assume that the empty set ∅ belongs to K. We refer to I ∈ K as an(abstract) simplex of K.

One-element simplices are called vertices of K. If K contains all one-elementsubsets of V , then we say that K is a simplicial complex on the vertex set V .

It is sometimes convenient to consider simplicial complexes K whose vertex setsare proper subsets of V . In this case we refer to a one-element subset of V which isnot a vertex of K as a ghost vertex .

The dimension of a simplex I ∈ K is dim I = |I| − 1, where |I| denotes thenumber of elements in I. The dimension of K is the maximal dimension of itssimplices. A simplicial complex K is pure if all its maximal simplices have the samedimension. A subcollection K′ ⊂ K which is itself a simplicial complex is called asubcomplex of K.

A geometric simplicial complex P is said to be a geometric realisation of anabstract simplicial complex K on a set V if there is a bijection between V and thevertex set of P which maps abstract simplices of K to vertex sets of faces of P .


Both geometric and abstract simplicial complexes will be assumed to be finite,unless we explicitly specify otherwise. In most constructions we identify the setV with the index set [m] = 1, . . . ,m and consider abstract simplicial complexeson [m]. An identification of V with [m] fixes an order of vertices, although thisorder will be irrelevant in most cases. We drop the parentheses in the notation ofone-element subsets i ⊂ [m], so that i ∈ K means that i is a vertex of K, andi /∈ K means that i is a ghost vertex.

We shall use the common notation ∆m−1 for any of the following three objects:an (m−1)-simplex (a convex polytope), the geometric simplicial complex consistingof all faces in an (m− 1)-simplex, and the abstract simplicial complex consisting ofall subsets of [m].

Construction 2.2.3. Every abstract simplicial complex K on [m] can be re-alised geometrically in Rm as follows. Let e1, . . . , em be the standard basis in Rm,and for each I ⊂ [m] denote by ∆I the convex hull of points ei with i ∈ I. Then

⋃

I∈K

∆I ⊂ Rm

is a geometric realisation of K.

The above construction is just a geometrical interpretation of the fact that anysimplicial complex on [m] is a subcomplex of the simplex ∆m−1. Also, by a clas-sical result [263], a d-dimensional abstract simplicial complex admits a geometricrealisation in R2d+1.

Example 2.2.4. The boundary of a simplicial n-polytope is a simplicial com-plex of dimension n − 1. For a simple polytope P , we shall denote by KP theboundary complex ∂P ∗ of the dual polytope. It coincides with the nerve of thecovering of ∂P by the facets. That is, the vertices of KP are the facets of P , and aset of vertices spans a simplex whenever the intersection of the corresponding facetsis nonempty. We refer to KP as the nerve complex of P .

Definition 2.2.5. The f -vector of an (n− 1)-dimensional simplicial complexK is f (K) = (f0, f1, . . . , fn−1), where fi is the number of i-dimensional simplicesin K. We also set f−1 = 1; to justify this convention one can assign dimension−1 to the empty simplex. The h-vector h(K) = (h0, h1, . . . , hn) is defined by theidentity

(2.3) h0sn + h1s

n−1 + · · ·+ hn = (s− 1)n + f0(s− 1)n−1 + · · ·+ fn−1.

(Warning: this is not the identity obtained by substituting t = 1 in (1.9).) Theg-vector g(K) = (g0, g1, . . . , g[n/2]) is defined by g0 = 1 and gi = hi − hi−1 fori = 1, . . . , [n/2].

Remark. If K = KP is the nerve complex of a simple polytope P , then f (K) =f (P ∗) and h(K) = h(P ). This our notational convention may look artificial, but itseems to be the best possible way to treat the face vectors of both polytopes andsimplicial complexes consistently.

Definition 2.2.6. Let K1, K2 be simplicial complexes on the sets [m1], [m2]respectively, and P1, P2 their geometric realisations. A map ϕ : [m1]→ [m2] inducesa simplicial map between K1 and K2 if ϕ(I) ∈ K2 for any I ∈ K1. A simplicial mapϕ is said to be nondegenerate if |ϕ(I)| = |I| for any I ∈ K1. On the geometric level,


a simplicial map extends linearly on the faces of P1 to a map P1 → P2, which wecontinue to denote by ϕ. A simplicial isomorphism is a simplicial map for whichthere exists a simplicial inverse.

There is an obvious isomorphism between any two geometric realisations of anabstract simplicial complex K. We therefore shall use the common notation |K| forany geometric realisation of K. Whenever it is safe, we shall not distinguish betweenabstract simplicial complexes and their geometric realisations. For example, weshall say ‘simplicial complex K is homeomorphic to X ’ instead of ‘the geometricrealisation of K is homeomorphic to X ’.

We shall refer to a simplicial complex homeomorphic to a topological space Xas a triangulation of X , or a simplicial subdivision of X .

A subdivision of a geometric simplicial complex P is a geometric simplicialcomplex P ′ such that each simplex of P ′ is contained in a simplex of P and eachsimplex of P is a union of finitely many simplices of P ′. A PL map ϕ : P1 →P2 is a simplicial map from a subdivision of P1 to a subdivision of P2. A PLhomeomorphism is a PL map for which there exists a PL inverse. In other words,two simplicial complexes are PL homeomorphic if there is a simplicial complexisomorphic to a subdivision of each of them.

Remark. For a topological approach to PL maps (where a PL map is definedbetween spaces rather than their triangulations) we refer to standard sources onPL topology, such as [165] and [277].

Example 2.2.7.1. If P is a simple n-polytope then the nerve complex KP (see Example 2.2.4)

is a triangulation of an (n− 1)-dimensional sphere Sn−1.2. Let Σ be a simplicial fan in Rn withm edges generated by vectors a1, . . . ,am.

Its underlying simplicial complex is defined by

KΣ =i1, . . . , ik ⊂ [m] : a i1 , . . . ,aik span a cone of Σ

.

Informally, KΣ may be viewed as the intersection of Σ with a unit sphere. The fanΣ is complete if and only if KΣ is a triangulation of Sn−1. If Σ is a normal fan ofa simple n-polytope P , then KΣ = KP .

Construction 2.2.8 (join). Let K1 and K2 be simplicial complexes on setsV1 and V2 respectively. The join of K1 and K2 is the simplicial complex

K1 ∗ K2 =I ⊂ V1 ⊔ V2 : I = I1 ∪ I2, I1 ∈ K1, I2 ∈ K2

on the set V1 ⊔ V2. The join operation is associative by inspection.

Example 2.2.9.1. If K1 = ∆m1−1, K2 = ∆m2−1, then K1 ∗ K2 = ∆m1+m2−1.2. The simplicial complex ∆0 ∗K (the join of K and a point) is called the cone

over K and denoted coneK.3. Let S0 be a pair of disjoint points (a 0-sphere). Then S0 ∗ K is called the

suspension of K and denoted ΣK. Geometric realisations of coneK and ΣK are thetopological cone and suspension over |K| respectively.

4. Let P1 and P2 be simple polytopes. Then

KP1×P2 = KP1 ∗ KP2 .

(see Construction 1.1.11).


Construction 2.2.10. The fact that the product of two simplices is not asimplex makes triangulations of products of spaces more subtle. There is the fol-lowing canonical way to triangulate the product of two simplicial complexes whosevertices are ordered. Let K1 and K2 be simplicial complexes on [m1] and [m2]respectively (this is one of the few constructions where the order of vertices is im-portant: here it is an additional structure). We construct a simplicial complexK1 × K2 on [m1] × [m2] as follows. By definition, simplices of K1 × K2 are thosesubsets in products I1 × I2 of I1 ∈ K1 and I2 ∈ K2 which establish non-decreasingrelations between I1 and I2. More formally,

K1 × K2 =I ⊂ I1 × I2 : I1 ∈ K1, I2 ∈ K2,

and i 6 i′ implies j 6 j′ for any two pairs (i, j), (i′, j′) ∈ I.

We leave it as an exercise to check that |K1×K2| defines a triangulation of |K1|×|K2|.Note that K1 × K2 6= K2 × K1 in general. Note also that if K1 = K2 = K, then thediagonal is naturally a subcomplex in the triangulation of |K| × |K|.

Construction 2.2.11 (connected sum of simplicial complexes). Let K1 andK2 be two pure d-dimensional simplicial complexes on sets V1 and V2 respectively,where |V1| = m1, |V2| = m2. Choose two maximal simplices I1 ∈ K1, I2 ∈ K2 andfix an identification of I1 with I2. Let V1 ∪I V2 be the union of V1 and V2 in whichI1 is identified with I2, and denote by I the subset created by the identification.We have |V1 ∪I V2| = m1 + m2 − d − 1. Both K1 and K2 now can be viewed assimplicial complexes on the set V1 ∪I V2. We define the connected sum of K1 andK2 at I1 and I2 as the simplicial complex

K1 #I1,I2 K2 = (K1 ∪ K2) \ Ion the set V1 ∪I V2. When the choices are clear, or their effect on the resultirrelevant, we use the abbreviation K1 # K2. Geometrically, the connected sumof |K1| and |K2| is produced by attaching |K1| to |K2| along I1 and I2 and thenremoving the face I obtained by the identification.

Example 2.2.12. Connected sum of simple polytopes defined in Construc-tion 1.1.13 is dual to the operation described above. Namely, if P and Q are twosimple n-polytopes, then

KP #KQ = KP#Q.

Definition 2.2.13. Let K be a simplicial complex on a set V . The link andthe star of a simplex I ∈ K are the subcomplexes

lkK I =J ∈ K : I ∪ J ∈ K, I ∩ J = ∅

;

stK I =J ∈ K : I ∪ J ∈ K

.

We also define the subcomplex

∂ stK I =J ∈ K : I ∪ J ∈ K, I 6⊂ J

.

Then we have a sequence of inclusions

lkK I ⊂ ∂ stK I ⊂ stK I.

For any vertex v ∈ K, the subcomplex stK v is the cone over lkK v = ∂ stK v. Also,| stK v| is the union of all faces of |K| that contain v. We omit the subscript K inthe notation of link and star whenever the ambient simplicial complex is clear.


The links of simplices determine the topological structure of the space |K|near any of its points. In particular, the following proposition describes the ‘localcohomology’ of |K|.

Proposition 2.2.14. Let x be an interior point of a simplex I ∈ K. Then

Hi(|K|, |K| \ x

) ∼= Hi−|I|(lk I),

where Hi(X,A) denotes the ith relative singular cohomology group of a pair A ⊂ X,

and Hi(K) denotes the ith reduced simplicial cohomology group of K.

Proof. We have

Hi(|K|, |K| \ x

) ∼= Hi(st I, (st I) \ x

) ∼= Hi(st I, (∂I) ∗ (lk I)

) ∼=∼= Hi−1

((∂I) ∗ (lk I)

) ∼= Hi−|I|(lk I).

Here the first isomorphism follows from the excision property, the second uses thefact that (∂I)∗ (lk I) is a deformation retract of (st I)\x, the third follows from thehomology sequence of pair, and the fourth is by the suspension isomorphism.

Given a subcomplex L ⊂ K, define the closed combinatorial neighbourhood ofL in K by

UK(L) =⋃

I∈L

stK I.

That is, UK(L) consists of all simplices of K, together with all their faces, havingsome simplex of L as a face. Define also the open combinatorial neighbourhood

UK(L) of |L| in |K| as the union of relative interiors of simplices of |K| having somesimplex of |L| as a face.

Definition 2.2.15. Given a subset I ⊂ V , define the corresponding full sub-complex of K (or the restriction of K to I) as

(2.4) KI =J ∈ K : J ⊂ I

.

Set coreV = v ∈ V : st v 6= K. The core of K is the subcomplex coreK = KcoreV .Then we may write K = core(K) ∗ ∆s−1, where ∆s−1 is the simplex on the setV \ coreV .

Example 2.2.16.1. lkK ∅ = K.2. Let K = ∂∆3 be the boundary of the tetrahedron on four vertices 1, 2, 3, 4,

and I = 1, 2. Then lk I consists of two disjoint points 3 and 4.3. Let K be the cone over L with vertex v. Then lk v = L, st v = K, and

coreK = coreL.

Exercises.

2.2.17. Assume that K1 is realised geometrically in Rn1 and K2 in Rn2 . Con-struct a realisation of the join K1 ∗ K2 in Rn1+n2+1.

2.2.18. Show that |K1 × K2| is a triangulation of |K1| × |K2|.2.2.19. Let K be a pure simplicial complex. Then lk I is pure of dimension

dimK − |I| for any I ∈ K.


2.3. Barycentric subdivision and flag complexes

Definition 2.3.1. The barycentric subdivision of an abstract simplicial com-plex K is the simplicial complex K′ defined as follows. The vertex set of K′ is theset I ∈ K, I 6= ∅ of nonempty simplices of K. Simplices of K′ are chains of em-bedded simplices of K. That is, I1, . . . , Ir ∈ K′ if and only if I1 ⊂ I2 ⊂ · · · ⊂ Irin K (after possible reordering) and I1 6= ∅.

The barycentre of a geometric simplex in Rd with vertices v1, . . . , vd+1 is thepoint 1

d+1(v1 + · · · + vd+1). A geometric realisation of K′ may be obtained by

mapping every vertex of K′ to the barycentre of the corresponding simplex of |K|;simplices of |K′| are therefore spanned by the sets of barycentres of chains of em-bedded simplices of |K|.

Example 2.3.2. For any (n−1)-dimensional simplicial complex K on [m], thereis a nondegenerate simplicial map K′ → ∆n−1 defined on the vertices by I 7→ |I|for I ∈ K, where |I| denotes the cardinality of I. Here I is viewed as a vertex of K′

and |I| as a vertex of ∆n−1.

Example 2.3.3. Let K be a simplicial complex on a set V , and assume weare given a choice function c : K → V assigning to each simplex I ∈ K one of itsvertices. For instance, if V = [m] we can define c(I) as the minimal element of I.For every such f there is a canonical simplicial map ϕc : K′ → K constructed asfollows. We define ϕc on the vertices of K′ by ϕc(I) = c(I) and then extend it tosimplices of K′ by the formula

ϕc(I1 ⊂ I2 ⊂ · · · ⊂ Ir) = c(I1), c(I2), . . . , c(Ir).The right hand side is a subset of Ir, and therefore it is a simplex of K.

We shall need explicit formulae for the transformation of the f - and h-vectorsunder the barycentric subdivision. Introduce the matrix

B = (bij), 0 6 i, j 6 n− 1; bij =

i∑

k=0

(−1)k(i+ 1

k

)(i− k + 1)j+1.

It can be shown that bij = 0 for i > j and bii = (i+1)!, so that B is a nonsingularupper triangular matrix.

Lemma 2.3.4. Let K′ be the barycentric subdivision of an (n− 1)-dimensionalsimplicial complex K. Then the f -vectors of K and K′ are related by the identity

fi(K′) =n−1∑

j=i

bijfj(K), 0 6 i 6 n− 1,

that is, f t(K′) = Bf t(K) where f t(K) is the column vector with entries fi(K).Proof. Consider the barycentric subdivision of a j-simplex ∆j , and let b′ij be

the number of its i-simplices which lie inside ∆j , i.e. not contained in ∂∆j . Then

we have fi(K′) =∑n−1

j=i b′ijfj(K). It remains to show that bij = b′ij . Indeed, it is

easy to see that the numbers b′ij satisfy the following recurrence relation:

b′ij = (j + 1)b′i−1,j−1 +(j+12

)b′i−1,j−2 + · · ·+

(j+1j−i+1

)b′i−1,i−1.

It follows by induction that b′ij is given by the same formula as bij .

2.3. BARYCENTRIC SUBDIVISION AND FLAG COMPLEXES 71

Now introduce the matrix

D = (dpq), 0 6 p, q 6 n; dpq =

p∑

k=0

(−1)k(n+ 1

k

)(p− k)q(p− k + 1)n−q,

where we set 00 = 1.

Lemma 2.3.5. The h-vectors of K and K′ are related by the identity:

hp(K′) =

n∑

q=0

dpqhq(K), 0 6 p 6 n,

that is, ht(K′) = Dht(K). Moreover, the matrix D is nonsingular.

Proof. The formula for hp(K′) is established by a routine check usingLemma 2.3.4, relations (2.3) and identities for the binomial coefficients. If we addthe component f−1 = 1 to the f -vector and change the matrix B appropriately,then we obtain D = C−1BC, where C is the transition matrix from the h-vectorto the f -vector (its explicit form can be obtained easily from relations (2.3)). Thisimplies the nonsingularity of D.

Definition 2.3.6. Let P be a poset (partially ordered set) with strict orderrelation <. Its order complex ord(P) is the collection of all totally ordered chainsx1 < x2 < · · · < xk (or flags), xi ∈ P . Clearly, ord(P) is a simplicial complex.

The following proposition is clear from the definition.

Proposition 2.3.7. Let K be a simplicial complex, viewed as the poset of itssimplices with respect to inclusion. Then ord(K \ ∅) is the barycentric subdivi-sion K′. The order complex of K (with the empty simplex included) is cone(K′).

This observation may be used to define the barycentric subdivision of othercombinatorial objects. For example, let Q be a convex polytope, and Q the poset ofits proper faces. Then ord(Q) is a simplicial complex; moreover, it is the boundarycomplex of a simplicial polytope Q′ (an exercise), called the barycentric subdivisionof Q. The vertices of Q′ correspond to the barycentres of proper faces of Q.

Proposition 2.3.8. Let P be a simple polytope and let K = KP be its nervecomplex. Given a facet F ⊂ P , let v be the corresponding vertex of K. Then stK′ vis a triangulation of F .

Proof. We identify ∂P with K′ by mapping the barycentre of each properface of P to the corresponding vertex of K′. Under this identification, F is mappedto the union of simplices of K′ corresponding to chains G1 ⊂ · · · ⊂ Gk = F of facesof P ending at F . This union is exactly the star of v in K′.

Definition 2.3.9. A simplicial complex K is called a flag complex if any setof vertices of K which are pairwise connected by edges spans a simplex.

A flag complex is therefore completely determined by its 1-skeleton, which isa simple graph. Given such a graph Γ, we may reconstruct the flag complex KΓ,whose simplices are the vertex sets of complete subgraphs (or cliques) of Γ.

Flag complexes may be characterised in terms of their missing faces. A missingface of K is a subset I ⊂ [m] such that I /∈ K, but every proper subset of I is asimplex of K. Then K is flag if and only if each of its missing faces has two vertices.


Therefore, P is a flag simple n-polytope if and only if any set of its facets whosepairwise intersections are nonempty intersects at a face.

Example 2.3.10.1. The order complex of a poset is a flag complex.2. A simple polytope P is a flag polytope (see Definition 1.6.1) if and only if

its nerve complex KP is a flag complex.3. The boundary of a 5-gon is a flag complex, but it is not the order complex

of a poset.4. The boundary of a d-simplex is not flag for d > 1.5. The join K1 ∗ K2 of two flag complexes (see Construction 2.2.8) is flag (an

exercise). Therefore, the product P ×Q of two flag simple polytopes is flag.6. The connected sum K1 #K2 of two d-dimensional complexes (see Construc-

tion 2.2.11) is not flag if d > 1.

Since the h-vector of a sphere triangulation is symmetric, the γ-vector γ(K) =(γ0, γ1, . . . , γ[n/2]) of an (n− 1)-dimensional sphere triangulation K can be definedby the equation

(2.5)n∑

i=0

hisitn−i =

[n/2]∑

i=0

γi(s+ t)n−2i(st)i.

Conjecture 2.3.11 (Gal [123]). Let K be a flag triangulation of an (n − 1)-sphere. Then γi(K) > 0 for i = 0, . . . ,

[n2

].

Substituting n = 2q, s = 1 and t = −1 into (2.5) we obtain

2q∑

i=0

(−1)ihi = (−1)qγq .

The top inequality γq > 0 from the Gal conjecture therefore implies the following:

Conjecture 2.3.12 (Charney–Davis [69]). The inequality

(−1)q2q∑

i=0

(−1)ihi(K) > 0

holds for any flag triangulation K of a (2q − 1)-sphere.

The Charney–Davis conjecture is a discrete analogue of the well-known Hopfconjecture of differential geometry, which states that the Euler characteristic χ ofa closed nonpositively curved Riemannian manifold M2q satisfies the inequality(−1)qχ > 0. Due to a theorem of Gromov [136], a piecewise Euclidean cubicalcomplex satisfies the discrete analogue of the nonpositive curvature condition, theso-called CAT(0) inequality, if and only if the links of all vertices are flag complexes.As was observed in [69], the Hopf conjecture for piecewise Euclidean cubical man-ifolds translates into Conjecture 2.3.12.

The Hopf and Charney–Davis conjectures are valid for q = 1, 2. In [123] theGal conjecture was verified for n 6 5.

Exercises.

2.3.13. Show that the matrix B of Lemma 2.3.4 satisfies bij = 0 for i > j andbii = (i+ 1)!.

2.4. ALEXANDER DUALITY 73

2.3.14. Prove the formula for hp(K′) of Lemma 2.3.5.

2.3.15. Show that the order complex of the poset of proper faces of a polytopeis the boundary complex of a simplicial polytope. (Hint: use stellar subdivisions,see Section 2.7 and the exercises there.)

2.3.16. The join of two flag complexes is flag.

2.3.17. Links of simplices in a flag complex are flag.

2.4. Alexander duality

For any simplicial complex, a dual complex may be defined on the same set.This duality has many important combinatorial and topological consequences.

Definition 2.4.1 (dual complex). Let K be a simplicial complex on [m] andK 6= ∆m−1. Define

K =I ⊂ [m] : [m] \ I /∈ K

.

Then K is also a simplicial complex on [m], which we refer to as the Alexander dual

of K. Obviously, the dual of K is K.

Construction 2.4.2. The barycentric subdivisions of both K and K can berealised as subcomplexes in the barycentric subdivision of the boundary of thestandard simplex on the set [m] in the following way.

A face of (∂∆m−1)′ corresponds to a chain I1 ⊂ · · · ⊂ Ir of included subsetsin [m] with I1 6= ∅ and Ir 6= [m]. We denote this face by ∆I1⊂···⊂Ir . (For example,∆i is the ith vertex of ∆m−1 regarded as a vertex of (∂∆m−1)′.) Then

(2.6) G(K) =⋃

I1⊂···⊂Ir, Ir∈K

∆I1⊂···⊂Ir

is a geometric realisation of K′. Denote i = [m]\i and, more generally, I = [m]\Ifor any subset I ⊂ [m]. Define the following subcomplex in (∂∆m−1)′:

D(K) =⋃

I1⊂···⊂Ir , Ir /∈K

∆Ir⊂···⊂I1.

Proposition 2.4.3. For any simplicial complex K 6= ∆m−1 on the set [m],D(K) is a geometric realisation of the barycentric subdivision of the dual complex:

(2.7) D(K) = |K′|.Moreover, the open combinatorial neighbourhood of complex (2.6) realising K′ in

(∂∆m−1)′ coincides with the complement of the complex D(K) realising K′:

U (∂∆m−1)′(|K′|

)=(∂∆m−1

)′ \ |K′|.In particular, |K′| is a deformation retract of the complement to |K′| in (∂∆m−1)′.

Proof. We map a vertex i of K to the vertex i = ∆i of (∂∆m−1)′, and map

the barycentre of a face I ∈ K to the vertex ∆I . Then the whole complex K′ ismapped to the subcomplex

⋃

I1⊂···⊂Ir, Ir∈K

∆Ir⊂···⊂I1,

which is the same as D(K). The second statement is left as an exercise.


Example 2.4.4. Let K be the boundary of a 4-gon with vertices 1, 2, 3, 4 (see

Fig. 2.2). Then K consists of two disjoint segments. The picture shows both K′

and K′ as subcomplexes in (∂∆3)′.

t1

t2

t3

t4

❩❩❩

❩❩❩❩❩❩❩❩❩

❩❩❩❩❩❩❩❩❩❩❩❩

❩❩❩❩❩❩❩❩❩❩❩❩

❩❩❩❩❩❩❩❩❩❩❩❩

❩❩❩❩❩❩❩❩❩❩❩❩

❩❩❩❩❩❩❩❩❩❩❩❩

❩❩❩❩❩❩❩❩❩❩❩❩

❩❩❩❩❩❩❩❩❩❩❩❩

❩❩❩❩❩❩❩

❩❩❩❩❩

3 1

❳❳❳❳❳❳ ❳❳❳❳❳❳

2

4

|K′|

|K′|

Figure 2.2. Dual complex and Alexander duality.

Theorem 2.4.5 (Combinatorial Alexander duality). For any simplicial complexK 6= ∆m−1 on the set [m] there is an isomorphism

Hj(K) ∼= Hm−3−j(K), for − 1 6 j 6 m− 2,

where Hk(·) and Hk(·) denote the kth reduced simplicial homology and cohomol-

ogy group (with integer coefficients) respectively. Here we assume that H−1(∅) =

H−1(∅) = Z.

Proof. Since (∂∆m−1)′ is homeomorphic to Sm−2, the Alexander duality the-orem [151, Theorem 3.44] and Proposition 2.4.3 imply that

Hj(K) = Hj(

U (∂∆m−1)′(|K′|))= Hj

((∂∆m−1)′ \ |K′|

)

= Hj(Sm−2 \ |K|

) ∼= Hm−3−j(K).

A more direct topological proof is outlined in Exercise 2.4.10. Theorem 2.4.5can be also proved in a purely combinatorial way, see [32]. There is also a proofwithin ‘combinatorial commutative algebra’ (which is the subject of Chapter 3),see Exercise 3.2.15 or [225, Theorem 5.6].

The duality between K and K extends to a duality between full subcomplexes

of K and links of simplices in K:

2.4. ALEXANDER DUALITY 75

Corollary 2.4.6. Let K 6= ∆m−1 be a simplicial complex on [m] and I /∈ K,

that is, I ∈ K. Then there is an isomorphism

Hj(KI) ∼= H|I|−3−j

(lkK I

), for − 1 6 j 6 |I| − 2.

Proof. We apply Theorem 2.4.5 to the complex KI , viewed as a simplicialcomplex on the set I of |I| elements. It follows from the definition that the dual

complex is lkK I, which also can be viewed as a simplicial complex on the set I.

Example 2.4.7. Let K be the boundary of a pentagon. Then K is a Mobius

band triangulated as shown on Fig. 2.3. Note that this K can be realised as asubcomplex in ∂∆4, and therefore it can be realised in R3 as a subcomplex in theSchlegel diagram of ∆4, see Construction 2.5.2.

r

rrr

r r r r r

r r r

❯

1

2

3

4

51 2 3 4

4 5 1

K K

Figure 2.3. The boundary of a pentagon and its dual complex.

Adding a ghost vertex to K results in suspending K, up to homotopy. Theprecise statement is as follows (the proof is clear and is omitted).

Proposition 2.4.8. Let K be a simplicial complex on [m] and K be the complexon [m+1] obtained by adding one ghost vertex = m+1 to K. Then the maximal

simplices of K are [m] and I ∪ for I ∈ K, that is,

K = ∆m−1 ∪K cone K.In particular, K is homotopy equivalent to the suspension ΣK.

Exercises.

2.4.9. Show that

U (∂∆m−1)′(|K′|

)=(∂∆m−1

)′ \ |K′|.2.4.10. Complete the details in the following direct proof of combinatorial

Alexander duality (Theorem 2.4.5).There is a simplicial map

ϕ : (∂∆m−1)′ → K′ ∗ K′

which is constructed as follows. We identify |K′| with G(K) and |K′| with D(K),see (2.6) and (2.7). The vertex sets of these two subcomplexes split the vertex setof (∂∆m−1)′ into two nonintersecting subsets. Therefore, ϕ is uniquely determined


on the vertices. Check that ϕ is indeed a simplicial map. It induces a map ofsimplicial cochains

ϕ∗ : Cj(K′)⊗ Cm−3−j(K′)→ Cm−2((∂∆m−1)′

),

or, equivalently,

Cj(K′)→ Cm−3−j(K′)⊗ Cm−2((∂∆m−1)′

).

By evaluating on the fundamental cycle of (∂∆m−1)′ in Cm−2 and passing to sim-plicial (co)homology, we obtain the required isomorphism

Hj(K) ∼=−→ Hm−3−j(K).

2.5. Classes of triangulated spheres

Boundary complexes of simplicial polytopes form an important albeit restrictedclass of triangulated spheres. In this section we review several other classes of trian-gulated spheres and related complexes, and discuss their role in topology, geometryand combinatorics.

Definition 2.5.1. A triangulated sphere (also known as a sphere triangulationor simplicial sphere) of dimension d is a simplicial complex K homeomorphic to ad-sphere Sd. A PL sphere is a triangulated sphere K which is PL homeomorphic tothe boundary of a simplex (equivalently, there exists a subdivision of K isomorphicto a subdivision of the boundary of a simplex).

Remark. A PL sphere is not the same as a ‘PL manifold homeomorphic toa sphere’, but rather a ‘PL manifold which is PL homeomorphic to the standardsphere’, where the standard PL structure on a sphere is defined by triangulating itas the boundary of a simplex. Nevertheless, the two notions coincide in dimensionsother than 4, see the discussion in the next section.

In small dimensions, triangulated spheres can be effectively visualised usingSchlegel diagrams and their generalisations, which we describe below.

Definition 2.5.2. By analogy with the notion of a geometric simplicial com-plex, we define a polyhedral complex as a collection C of convex polytopes in aspace Rn such that every face of a polytope in C belongs to C and the intersectionof any two polytopes in C is either empty or a face of each. Two polyhedral com-plexes C1 and C2 are said to be combinatorially equivalent if there is a one-to-onecorrespondence between their polytopes respecting the inclusion of faces.

The boundary ∂P of a convex polytope P is a polyhedral complex. Anotherimportant polyhedral complex associated with P can be constructed as follows.Choose a facet F of P and a point p /∈ P ‘close enough’ to F so that any segmentconnecting p to a point in P \ F intersects the relative interior of F (see Fig. 2.4for the case P = I3). Now project the complex ∂P onto F from the point p.The projection images of faces of P different from F form a polyhedral complex Csubdividing F . We refer to C, and also to any polyhedral complex combinatoriallyequivalent to it, as a Schlegel diagram of the polytope P .

An n-diagram is a polyhedral complex C consisting of n-polytopes and theirfaces and satisfying the following conditions:

(a) the union of all polytopes in C is an n-polytope Q;

2.5. CLASSES OF TRIANGULATED SPHERES 77

p

Figure 2.4. Schlegel diagram of I3.

(b) every nonempty intersection of a polytope from C with the boundary ofQ belongs to C.

We refer to Q as the base of the n-diagram C. An n-diagram is simplicial if itconsists of simplices and its base is a simplex.

By definition, a Schlegel diagram of an n-polytope is an (n− 1)-diagram.

Proposition 2.5.3. Let C be a simplicial (n − 1)-diagram with base Q. ThenC ∪∂Q Q is an (n− 1)-dimensional PL sphere.

Proof. We need to construct a simplicial complex which is a subdivision ofboth C ∪∂Q Q and ∂∆n. This can be done as follows. Replace one of the facets of∂∆n by the diagram C. The resulting simplicial complex K is a subdivision of ∂∆n.On the other hand, K is isomorphic to the subdivision of C ∪∂Q Q obtained byreplacing Q by a Schlegel diagram of ∆n.

Corollary 2.5.4. The boundary of a simplicial n-polytope is an (n − 1)-dimensional PL sphere.

In dimensions n 6 3 every (n − 1)-diagram is a Schlegel diagram of an n-polytope. Indeed, for n 6 2 this is obvious, and for n = 3 this is one of equivalentformulations of the well-known Steinitz Theorem (see [325, Theorem 5.8]). The firstexample of a 3-diagram which is not a Schlegel diagram of a 4-polytope was found byGrunbaum ([139, §11.5], see also [140]) as a correction of Bruckner’s result of 1909on the classification of simplicial 4-polytopes with 8 vertices. Another example wasfound by Barnette [18]. We describe Barnette’s example below. For the originalexample of Grunbaum, see Exercise 2.5.15.

Construction 2.5.5 (Barnette’s 3-diagram). Here a certain simplicial 3-diagram C will be constructed. Consider the octahedron Q obtained by twistingthe top face (abc) of a triangular prism (Fig. 2.5 (a)) slightly so that the verticesa, b, c, d, e and f are in general position. Assume that the edges (bd), (ce) and


d e

f

a b

c

(a)

d

e

f

a

b

c

p′

(b)

Figure 2.5. Barnette’s 3-diagram.

(af) lie inside Q. The tetrahedra (abde), (bcef) and (acdf) will be included in thecomplex C. Each of these tetrahedra has two faces which lie inside Q. These 6 tri-angles together with (abc) and (def) form a triangulated 2-sphere inside Q, whichwe denote by S. Now place a point p inside S so that the line segments from p tothe vertices of S lie inside S. We add to C the eight tetrahedra obtained by takingcones with vertex p over the faces of S (namely, the tetrahedra (pabc), (pdef),(pabd), (pbed), (pbce), (pcef), (pacf) and (padf)). Note that S = lk p. Applyinga projective transformation if necessary, we may assume that there is a point p′

outside the octahedron Q with the property that the segments joining p′ with thevertices of Q lie outside Q (see Fig. 2.5 (b)). Finally, we add to C the tetrahedraobtained by taking cones with vertex p′ over the faces of Q other than the face (def)(there are 7 such tetrahedra: (p′abc), (p′abe), (p′ade), (p′acd), (p′cdf), (p′bcf) and(p′bef)). The union of all tetrahedra in C is the tetrahedron (p′def); hence, C is asimplicial 3-diagram. It has 8 vertices and 18 tetrahedra.

Proposition 2.5.6. The 3-diagram C from the previous construction is not aSchlegel diagram.

Proof. Suppose there is a polytope P whose Schlegel diagram is C. Since Pis simplicial, we may assume that its vertices are in general position. We label thevertices of P with the same letters as the corresponding vertices in C. Consider thecomplex S = lk p. Let P ′ be the convex hull of all vertices of P other than p. ThenP ′ is still a simplicial polytope, and S is a subcomplex in ∂P ′. In the complex ∂P ′

the sphere S is filled with tetrahedra whose vertices belong to S. Then at least oneedge of one of these tetrahedra lies inside S. However, any two vertices of S whichare not joined by an edge on S are joined by an edge of C lying outside S. Sincethe polytope P ′ cannot contain a double edge we have reached a contradiction.

2.5. CLASSES OF TRIANGULATED SPHERES 79

Now we can introduce two more classes of triangulated spheres.

Definition 2.5.7. A polytopal sphere is a triangulated sphere isomorphic tothe boundary complex of a simplicial polytope.

A starshaped sphere is a triangulated sphere isomorphic to the underlying com-plex of a complete simplicial fan. Equivalently, a triangulated sphere K of dimensionn− 1 is starshaped if there is a geometric realisation of K in Rn and a point p ∈ Rn

with the property that each ray emanating from p meets |K| in exactly one point.The set of points such p ∈ Rn is called the kernel of |K|.

Example 2.5.8. The triangulated 3-sphere coming from Barnette’s 3-diagramis known as the Barnette sphere. It is starshaped. Indeed, in the construction ofBarnette’s 3-diagram we have the octahedron Q ⊂ R3 and a vertex p′ outside Qsuch that lk p′ = ∂Q. If we choose the vertex p′ in R4 \ R3, then the Barnettesphere can be realised in R4 as the boundary complex of the pyramid with vertexp′ and base Q (subdivided as described in Construction 2.5.5). This realisation isobviously starshaped.

We therefore have the following hierarchy of triangulations:

(2.8)polytopalspheres

⊂ starshapedspheres

⊂ PLspheres

⊂ triangulatedspheres

Here the first inclusion follows from the construction of the normal fan (Construc-tion 2.1.3), and the second is left as an exercise.

In dimension 2 any triangulated sphere is polytopal (this is another corollaryof the Steinitz theorem). Also, by the result of Mani [205], any triangulated d-dimensional sphere with up to d + 4 vertices is polytopal. However, in general allinclusions above are strict.

The first inclusion in (2.8) is strict already in dimension 3, as is seen fromExample 2.5.8. There are 39 combinatorially different triangulations of a 3-spherewith 8 vertices, among which exactly two are nonpolytopal (namely, the Barnetteand Bruckner spheres); this classification was completed by Barnette [21].

The second inclusion in (2.8) is also strict in dimension 3. The first exampleof a nonstarshaped sphere triangulation was found by Ewald and Schulz [109]. Wesketch this example below, following [108, Theorem 5.5].

Example 2.5.9 (Nonstarshaped sphere triangulation). We use the fact, ob-served by Barnette, that not every tetrahedron in Barnette’s 3-diagram can bechosen as the base of a 3-diagram of the Barnette sphere (see Exercise 2.5.16). Forinstance, the tetrahedron (abcd) cannot be chosen as the base.

Now let K be a connected sum of two copies of the Barnette sphere along thetetrahedra (abcd) (the identification of vertices in the two tetrahedra is irrelevant).Assume that K has a starshaped realisation in R4. The hyperplane H through thepoints a, b, c, d splits |K| into two parts |K1| and |K2|. Since the kernel of |K| is anopen set, it contains a point p not lying in H . Then by projecting either |K1| or|K2| onto H from p, we obtain a 3-diagram of the Barnette sphere with base (abcd).This is a contradiction.

The fact that K is a PL sphere is an exercise.

The third inclusion in (2.8) is the subtlest one. It is known that in dimension 3any triangulated sphere is PL. In dimension 4 the corresponding question is open,


but starting from dimension 5 there exist non-PL sphere triangulations. See thediscussion in the next section and Example 2.6.3.

Many important open problems of combinatorial geometry arise from analysingthe relationships between different classes of sphere triangulations. We end thissection by discussing some of these problems.

In connection with the condition of realisability of a triangulated (n−1)-spherein Rn in the definition of a starshaped sphere, we note that the existence of such arealisation is open in general:

Problem 2.5.10. Does every PL (n− 1)-sphere admit a geometric realisationin an n-dimensional space?

The g-theorem (Theorem 1.4.14) gives a complete characterisation of integralvectors arising as the f -vectors of polytopal spheres. It is therefore natural toask whether the g-theorem extends to all sphere triangulations. This question wasposed by McMullen [218] as an extension of his conjecture for simplicial polytopes.Since 1980, when McMullen’s conjecture for simplicial polytopes was proved byBillera, Lee, and Stanley, its generalisation to spheres has been regarded as themain open problem in the theory of f -vectors:

Problem 2.5.11 (g-conjecture for triangulated spheres). Does Theorem 1.4.14hold for triangulated spheres?

The g-conjecture is open even for starshaped spheres. Note that only thenecessity of the conditions in the g-theorem (that is, the fact that every g-vector isan M -vector) has to be verified for triangulated spheres. If correct, the g-conjecturewould imply a complete characterisation of f -vectors of triangulated spheres.

The Dehn–Sommerville relations (condition (a) in Theorem 1.4.14) hold forarbitrary sphere triangulations (see Corollary 3.4.7 below). The f -vectors of trian-gulated spheres also satisfy the UBT and LBT inequalities given in Theorems 1.4.4and 1.4.9 respectively. The proof of the Lower Bound Theorem for simplicial poly-topes given by Barnette in [20] extends to all triangulated spheres (see also [177]).In particular, this implies the second GLBC inequality h1 6 h2, see (1.18). TheUpper Bound Theorem for triangulated spheres was proved by Stanley [288] (weshall give his argument in Section 3.3). This implies that the g-conjecture is truefor triangulated spheres of dimension 6 4. The third GLBC inequality h2 6 h3 (forspheres of dimension > 5) is open.

Many attempts to prove the g-conjecture were made after 1980. Though un-successful, these attempts resulted in some very interesting reformulations of theg-conjecture. The results of Pachner [253] reduce the g-conjecture (for PL spheres)to some properties of bistellar moves ; see the discussion after Theorem 2.7.3 below.

The lack of progress in proving the g-conjecture motivated Bjorner and Lutz tolaunch a computer-aided search for counterexamples [31]. Though their bistellarflip algorithm and BISTELLAR software produced many remarkable results ontriangulations of manifolds, no counterexamples to the g-conjecture were found.More information on the g-conjecture and related questions may be found in [293]and [325, Lecture 8].

We may extend the hierarchy (2.8) by considering polyhedral complexes homeo-morphic to spheres (the so-called polyhedral spheres) instead of triangulated spheres.There are obvious analogues of polytopal and PL spheres in this generality. How-ever, unlike the case of triangulations, the two definitions of a starshaped sphere

2.6. TRIANGULATED MANIFOLDS 81

(namely, the one using fans and the one using the kernel points) no longer pro-duce the same classes of objects, see [108, § III.5] for the corresponding examples.One of the most notorious and long standing problems is to find a proper higherdimensional analogue to the Steinitz theorem. This theorem characterises graphsof 3-dimensional polytopes, and one of its equivalent formulations is that everypolyhedral 2-sphere is polytopal. In higher dimensions, identification of the classof polytopal spheres inside all polyhedral spheres is known as the Steinitz problem:

Problem 2.5.12 (Steinitz Problem). Find necessary and sufficient conditionsfor a polyhedral decomposition of a sphere to be combinatorially equivalent to theboundary complex of a convex polytope.

This is far from being solved even in the case of triangulated spheres. For moreinformation on the relationships between different classes of polyhedral spheres andcomplexes see the above cited book of Ewald [108] and the survey article by Kleeand Kleinschmidt [187].

Exercises.

2.5.13. Show that K is the underlying complex of a complete simplicial fan ifand only if there is a geometric realisation of K in Rn and a point p ∈ Rn with theproperty that each ray emanating from p meets |K| in exactly one point.

2.5.14. Prove that every starshaped sphere is a PL sphere.

2.5.15 (Bruckner sphere). The Bruckner sphere is obtained by replacing twotetrahedra (pabc) and (p′abc) in the Barnette sphere by three tetrahedra (pp′ab),(pp′ac) and (pp′bc) (this is an example of a bistellar 1-move considered in Sec-tion 2.7). Show that the Bruckner sphere is starshaped but not polytopal. Notethat the 1-skeleton of the Bruckner sphere is a complete graph (that is, the Brucknersphere is a neighbourly triangulation, see Definition 1.1.15).

2.5.16. Show that the tetrahedron (abcd) in Barnette’s 3-diagram (Construc-tion 2.5.5) cannot be chosen as the base of a 3-diagram of the Barnette sphere.Which tetrahedra can be chosen as the base?

2.5.17. The connected sum of two PL spheres is a PL sphere.

2.6. Triangulated manifolds

Piecewise linear topology experienced an intensive development during the sec-ond half of the 20th century, thanks to the efforts of many topologists. Surgerytheory for simply connected manifolds of dimension > 5 originated from the earlywork of Milnor, Kervaire, Browder, Novikov and Wall, culminated in the proof of thetopological invariance of rational Pontryagin classes given by Novikov in 1965, andwas further developed in the work of Lashof, Rothenberg, Sullivan, Kirby, Sieben-mann, and others. It led to a better understanding of the place of PL manifoldsbetween the topological and smooth categories. Without attempting to overviewthe current state of the subject, which is generally beyond the scope of this book,we include several important results on the triangulation of topological manifolds,with a particular emphasis on various nonexamples. We also provide references forfurther reading.

All manifolds here are compact, connected and closed, unless otherwise stated.


Definition 2.6.1. A triangulated manifold (or simplicial manifold) is a sim-plicial complex K whose geometric realisation |K| is a topological manifold.

A PL manifold is a simplicial complex K of dimension d such that lk I is a PLsphere of dimension d− |I| for every nonempty simplex I ∈ K.

Every PLmanifold |K| of dimension d is a triangulated manifold: it has an atlaswhose change of coordinates functions are piecewise linear. Indeed, for each vertexv ∈ |K| the (d− 1)-dimensional PL sphere lk v bounds an open neighbourhood Uvwhich is homeomorphic to an open d-ball. Since any point of |K| is contained inUv for some v, this defines an atlas for |K|.

Remark. The term ‘PL manifold’ is often used for a manifold with a PLatlas, while its particular triangulation with the property above is referred to as acombinatorial manifold. We shall not distinguish between these two notions.

Does every triangulation of a topological manifold yield a simplicial complexwhich is a PL manifold? The answer is ‘no’, and the question itself ascends to afamous conjecture of the dawn of topology, known as the Hauptvermutung, whichis German for ‘main conjecture’. Below we briefly review the current status of thisconjecture, referring to the survey article [274] by Ranicki for a much more detailedhistorical account and more references.

In the early days of topology all of the known topological invariants were definedin combinatorial terms, and it was very important to find out whether the topol-ogy of a triangulated space fully determines the combinatorial equivalence class ofthe triangulation (in the sense of Definition 2.2.6). In the modern terminology,the Hauptvermutung states that any two homeomorphic simplicial complexes arecombinatorially equivalent (PL homeomorphic). This is valid in dimensions 6 3;the result is due to Rado (1926) for 2-manifolds, Papakyriakopoulos (1943) for 2-complexes, Moise (1953) for 3-manifolds, and E. Brown (1963) for 3-complexes [43];see [233] for a detailed exposition. The first examples of complexes disproving theHauptvermutung in dimensions > 6 were found by Milnor in the early 1960s. How-ever, the manifold Hauptvermutung, namely the question of whether two homeo-morphic triangulated manifolds are combinatorially equivalent, remained open untilthe end of the 1960s. The first counterexamples were found by Siebenmann in 1969,and relied heavily on the topological surgery theory. The ‘double suspension theo-rem’, which we state as Theorem 2.6.2 below, appeared around 1975 and providedmuch more explicit counterexamples to the manifold Hauptvermutung.

A d-dimensional homology sphere (or simply homology d-sphere) is a topologicald-manifold whose integral homology groups are isomorphic to those of a d-sphere Sd.

Theorem 2.6.2 (Edwards, Cannon). The double suspension of any homologyd-sphere is homeomorphic to Sd+2.

This theorem was proved for most double suspensions and all triple suspensionsby Edwards [103]; the general case was done by Cannon [65]. One of its most im-portant consequences is the existence of non-PL triangulations of 5-spheres, whichalso disproves the manifold Hauptvermutung in dimensions > 5.

Example 2.6.3 (non-PL triangulated 5-sphere). Let M be a triangulated ho-mology 3-sphere which is not homeomorphic to S3. An example of such M isprovided by the Poincare sphere. It is the homogeneous space SO(3)/A5, wherethe alternating group A5 is represented in R3 as the group of self-transformations

2.6. TRIANGULATED MANIFOLDS 83

of a dodecahedron. A particular symmetric triangulation of the Poincare sphereis given in [31]. By Theorem 2.6.2, the double suspension Σ2M is homeomorphicto S5 (and, more generally, ΣkM is homeomorphic to Sk+3 for k > 2). However,Σ2M cannot be a PL sphere, since M appears as the link of a 1-simplex in Σ2M .

Also, according to a result of Bjorner and Lutz [31], for any d > 5 there is anon-PL triangulation of Sd with d+ 13 vertices.

Theorem 2.6.2 led to progress in the following ‘manifold recognition problem’:given a simplicial complex, how one can decide whether its geometric realisation isa topological manifold? In higher dimensions there is the following result, whichcan be viewed as a generalisation of the double suspension theorem.

Theorem 2.6.4 (Edwards [104]). For d > 5 the realisation of a simplicial com-plex K is a topological manifold of dimension d if and only if lk I has the homologyof a (d− |I|)-sphere for each nonempty simplex I ∈ K, and lk v is simply connectedfor each vertex v of K.

Remark. From the algorithmic point of view, the homology of links is easilycomputable, but their simply connectedness seems to be undecidable. There is arelated result of Novikov [318, Appendix] that a triangulated 5-sphere cannot bealgorithmically recognised. On the other hand, the algorithmic recognition problemfor a triangulated 3-sphere has a positive solution, with the first algorithm providedby Rubinstein [278]. See detailed exposition in Matveev’s book [213], which alsocontains a proof that all 3-dimensional Haken manifolds can be recognised and fullyclassified algorithmically.

With the discovery of exotic smooth structures on 7-spheres by Milnor andthe disproval of the Hauptvermutung it had become important to understand bet-ter the relationship between PL and smooth structures on topological manifolds.Since a PL structure implies the existence of a particular sort of triangulation, therelated question of whether a topological manifold admits any triangulation (notnecessarily PL) had also become important.

Triangulations of 2-manifolds have been known from the early days of topology.A proof that any 3-manifold can be triangulated was obtained independently byMoise and Bing in the end of 1950s (the proof can be found in [233]). Since thelink of a vertex in a triangulated 3-sphere is a 2-sphere, and a 2-sphere is alwaysPL, all topological 3-manifolds are PL.

A smooth manifold of any dimension has a PL triangulation by a theoremof Whitney (a proof can be found in [235]). Moreover, in dimensions 6 3 everytopological manifold has a unique smooth structure, see [233] for a proof. All theseconsiderations show that in dimensions up to 3 the categories of topological, PLand smooth manifolds are equivalent.

The situation in dimension 4 is quite different. There exist topological 4-manifolds that do not admit a PL triangulation. An example is provided by Freed-man’s fake CP 2 [118, §8.3, §10.1], a topological manifold which is homeomorphic,but not diffeomorphic to the complex projective plane CP 2. This example alsoshows that the Hauptvermutung is false in dimension 4. Even worse, some topo-logical 4-manifolds do not admit any triangulation; an example is the topological4-manifold with the intersection form E8, see [3].


In dimension 4 the categories of PL and smooth manifolds agree, that is, thereis exactly one smooth structure on every PL manifold. However, the classifica-tion of PL (or equivalently, smooth) structures is wide open even for the simplesttopological manifolds. The most notable problem here is the following.

Problem 2.6.5. Is a PL (or smooth) structure on a 4-sphere unique?

In dimensions > 5 the PL structure on a topological sphere is unique (that is,a PL manifold which is homeomorphic to a sphere is a PL sphere).

For the discussion of the classification of PL structures on topological man-ifolds we refer to Ranicki’s survey [274], the original essay [185] by Kirby andSiebenmann, and a more recent survey by Rudyak [279].

2.7. Stellar subdivisions and bistellar moves

By a theorem of Alexander, a common subdivision of two PL homeomorphicPL manifolds can be obtained by iterating operations from a very simple and ex-plicit list, known as stellar subdivisions. An even more concrete iterative descriptionof PL homeomorphisms was obtained by Pachner [252], who introduced the no-tion of bistellar moves (in other terminology, bistellar flips or bistellar operations),generalising the 2- and 3-dimensional flips from low-dimensional topology. Theseoperations allow us to decompose a PL homeomorphism into a sequence of simple‘moves’ and thus provide a very convenient way to compute and handle topologicalinvariants of PL manifolds. Starting from a given PL triangulation, bistellar op-erations may be used to construct new triangulations with some good properties,such as ones that are symmetric or have a small number of vertices. On the otherhand, bistellar moves can be used to produce some nasty triangulations if we startfrom a non-PL triangulation. Both approaches were used in the work of Bjorner–Lutz [31] and Lutz [200] to find many interesting triangulations of low-dimensionalmanifolds. In our exposition of bistellar moves we follow the terminology of [200].

Bistellar moves also provide a combinatorial interpretation for algebraic flop op-erations for projective toric varieties and for certain surgery operations on moment-angle complexes and torus manifolds. Finally, bistellar moves may be used to definea metric on the space of PL triangulations of a given PL manifold, see [238].

Definition 2.7.1 (stellar subdivisions and bistellar moves). Let I ∈ K be anonempty simplex of a simplicial complex K. The stellar subdivision of K at I isobtained by replacing the star of I by the cone over its boundary:

ssI K = (K \ stK I) ∪ (cone ∂ stK I).

If dim I = 0 then ssI K = K. Otherwise the complex ssI K acquires an additionalvertex (the vertex of the cone) whose link is ∂ stK I. Two possible stellar subdivi-sions of a 2-dimensional complex are shown in Fig. 2.6.

Now let K be a triangulated manifold of dimension d. Assume that I ∈ K is a(d − j)-face such that the simplicial complex lkK I is the boundary of a j-simplexJ which is not a face of K. Then the operation bmI on K defined by

bmI K =(K \ (I ∗ ∂J)

)∪ (∂I ∗ J)

is called a bistellar j-move. Since I ∗ ∂J = stK I and ∂I ∗ J = stK J , where

K = bmI K, the bistellar j-move is the composition of a stellar subdivision at Iand the inverse stellar subdivision at J , which explains the term. In particular,

2.7. STELLAR SUBDIVISIONS AND BISTELLAR MOVES 85

sssI K

dim I = 1

PPPPP

sssI K

dim I = 2

Figure 2.6. Stellar subdivisions at a 2-simplex and at an edge.

the stellar subdivision ssI K is a common subdivision of K and K, so that K and Kare combinatorially equivalent. Note that a 0-move is the stellar subdivision at amaximal simplex (we assume that the boundary of a 0-simplex is empty).

Bistellar j-moves with i >[d2

]are also called reverse (d− j)-moves. A 0-move

adds a new vertex to the triangulation, a d-move (reverse 0-move) deletes a vertex,and all other bistellar moves do not change the number of vertices. The bistellarmoves in dimension 2 and 3 are shown in Figures 2.7 and 2.8. The bistellar 1-movein dimension 3 replaces two tetrahedra with a common face by 3 tetrahedra with acommon edge.

Two simplicial complexes are said to be bistellarly equivalent if one can betransformed to another by a finite sequence of bistellar moves.

1-move

PPPPP

0-move

2-move

Figure 2.7. Bistellar moves for q = 2.


2-move

1-move

3-move

0-move

Figure 2.8. Bistellar moves for q = 3.

We have seen that bistellar equivalence implies PL homeomorphism. The fol-lowing result shows that for PL manifolds the converse is also true.

Theorem 2.7.2 (Pachner [252, Theorem 1], [253, (5.5)]). Two PL manifoldsare bistellarly equivalent if and only if they are PL homeomorphic.

The behaviour of the face numbers of a triangulation under bistellar movesis easily controlled. It can be most effectively described in terms of the g-vector,gi(K) = hi(K) − hi−1(K), 0 < i 6

[d2

]:

Theorem 2.7.3 (Pachner [252]). If a triangulated d-manifold K is obtainedfrom K by a bistellar k-move, 0 6 k 6

[d−12

], then

gk+1(K) = gk+1(K) + 1;

gi(K) = gi(K) for all i 6= k + 1.

Furthermore, if d is even and K is obtained from K by a bistellar[d2

]-move, then

gi(K) = gi(K) for all i.

This theorem allows us to interpret the inequalities from the g-conjecture forPL spheres (see Theorem 1.4.14) in terms of the numbers of bistellar k-movesneeded to transform a given PL sphere to the boundary of a simplex. For instance,the inequality h1 6 h2, n > 4, is equivalent to the statement that the numberof 1-moves in the sequence of bistellar moves taking a given (n − 1)-dimensionalPL sphere to the boundary of an n-simplex is less than or equal to the number ofreverse 1-moves. (Note that the g-vector of ∂∆n is (1, 0, . . . , 0).)

2.8. SIMPLICIAL POSETS AND SIMPLICIAL CELL COMPLEXES 87

Remark. There is also a generalisation of Theorem 2.7.2 to PL manifolds withboundary, see [253, (6.3)].

In the case of polytopal sphere triangulations a stellar subdivision is related toanother familiar operation:

Proposition 2.7.4. Assume given a simple polytope P and a proper face G ⊂P . Let P ∗ be the dual simplicial polytope, KP = ∂P ∗ its nerve complex, and J ⊂ P ∗

the face dual to G. Then the stellar subdivision ssJ KP is the nerve complex of the

polytope P obtained by the face truncation at G.

Proof. This follows directly by comparing the face poset of P , described inConstruction 1.1.12, with that of ssJ KP .

Exercises.

2.7.5. The barycentric subdivision of K can be obtained as a sequence of stellarsubdivisions at all faces I ∈ K, starting from the maximal ones.

2.7.6. Deduce formulae for the transformation of the f -, h- and g-vector of Kunder a stellar subdivision. Deduce similar formulae for a bistellar move (the caseof the g-vector is Theorem 2.7.3).

2.8. Simplicial posets and simplicial cell complexes

Simplicial posets describe the combinatorial structures underlying ‘generalisedsimplicial complexes’ whose faces are still simplices, but two faces are allowed tointersect in any subcomplex of their boundary, rather than just in a single face.These are also known as ‘ideal triangulations’ in low-dimensional topology, or as‘simplicial cell complexes’.

Definition 2.8.1. A poset (partially ordered set) S with order relation 6 is

called simplicial if it has an initial element 0 and for each σ ∈ S the lower segment

[0, σ] = τ ∈ S : 0 6 τ 6 σis the face poset of a simplex. (The latter poset is also known as a Boolean lattice,and simplicial posets are sometimes called Boolean posets.) We assume all our

posets to be finite. The rank function | · | on S is defined by setting |σ| = k if [0, σ]is the face poset of a (k − 1)-dimensional simplex. The rank of S is the maximumof ranks of its elements, and the dimension of S is its rank minus one. A vertex ofS is an element of rank one. We assume that S has m vertices, denote the vertexset by V (S), and usually identify it with [m] = 1, . . . ,m. Similarly, we denote byV (σ) the vertex set of σ, that is the set of i with i 6 σ.

The face poset of a simplicial complex is a simplicial poset, but there aremany simplicial posets that do not arise in this way (see Example 2.8.2 below).We identify a simplicial complex with its face poset, thereby regarding simplicialcomplexes as particular cases of simplicial posets.

To each σ ∈ S we assign a geometric simplex ∆σ whose face poset is [0, σ], andglue these geometric simplices together according to the order relation in S. As aresult we get a regular cell complex in which the closure of each cell is identifiedwith a simplex preserving the face structure, and all attaching and characteristicmaps are inclusions (see [151, Appendix] for the terminology of cell complexes).


We call it the simplicial cell complex associated with S and denote its underlyingspace by |S|.

In the case when S is (the face poset of) a simplicial complex K the space |S|is the geometric realisation |K|.

Remark. Using a more formal categorical language, we consider the face cat-egory (S) whose objects are elements σ ∈ S and there is a morphism from σ to τwhenever σ 6 τ . Define a diagram (covariant functor) ∆S from (S) to topologicalspaces by sending σ ∈ S to the geometric simplex ∆σ and sending every morphismσ 6 τ to the inclusion ∆σ → ∆τ . Then we may write

|S| = colim∆S ,

where the colimit (or direct limit) is taken in the category of topological spaces.This is the first example of colimit construction over the face category (S). Manyother examples of this sort will appear later.

In most circumstances we shall not distinguish between simplicial posets andsimplicial cell complexes. We shall also sometimes refer to elements σ ∈ S assimplices or faces of S.

Example 2.8.2. Consider the simplicial cell complex obtained by attachingtwo d-dimensional simplices along their boundaries. Its corresponding simplicialposet is not the face poset of a simplicial complex if d > 0.

Three cellular subdivisions of a circle are shown in Fig. 2.9. The first is not asimplicial cell complex. The second is a simplicial cell complex, but not a simplicialcomplex (it corresponds to d = 1 in the previous paragraph). The third one is asimplicial complex.

s

(1)

s s

(2)

ss s

(3)

Figure 2.9. Cellular subdivisions of a circle.

Construction 2.8.3 (folding a simplicial poset onto a simplicial complex).For every simplicial poset S there is the associated simplicial complex KS on thesame vertex set V (S), whose simplices are sets V (σ), σ ∈ S. There is a foldingmap of simplicial posets

(2.9) S −→ KS , σ 7→ V (σ).

It is identical on the vertices, and every simplex in KS gets covered by a finitenumber of simplices of S.

For any two elements σ, τ ∈ S, denote by σ ∨ τ the set of their least commonupper bounds (joins), and denote by σ ∧ τ the set of their greatest common lowerbounds (meets). Since S is a simplicial poset, σ ∧ τ consists of a single elementwhenever σ ∨ τ is nonempty. It is easy to observe that S is a simplicial complex

2.9. CUBICAL COMPLEXES 89

if and only if for any σ, τ ∈ S the set σ ∨ τ is either empty or consists of a singleelement. In this case S coincides with KS .

Applying barycentric subdivision to every simplex σ ∈ S we obtain a newsimplicial cell complex S ′, called the barycentric subdivision of S. From Proposi-tion 2.3.7 it is clear that S ′ can be identified with the (geometric realisation of the)

order complex ord(S \ 0). We therefore obtain the following.

Proposition 2.8.4. The barycentric subdivision S ′ of a simplicial cell complexis a simplicial complex.

Exercises.

2.8.5. Show that the following conditions are equivalent for a simplicial poset S:

(a) S is (the face poset of) a simplicial complex;(b) for any σ, τ ∈ S the set σ ∧ τ consists of a single element;(c) for any σ, τ ∈ S the set σ∨τ is either empty or consists of a single element.

2.9. Cubical complexes

At some stage of the development of combinatorial topology, cubical complexeswere considered as an alternative to triangulations, a new way to study topologicalinvariants combinatorially. Their nice feature is that the product of cubes is again acube, which makes subdivisions of products easier and leads to a more straightfor-ward definition of the multiplication in cohomology. Later it turned out, however,that the cubical (co)homomology itself is not particularly advantageous in compar-ison with the simplicial one. Currently combinatorial geometry is the main field ofapplications of cubical complexes; moreover, subdivisions into cubes sometimes arevery helpful in various geometrical and topological considerations. In this sectionwe collect the necessary definitions and notation, and then proceed to describe someimportant cubical decompositions of simple polytopes and simplical complexes.

Definitions and examples. As in the case of simplicial complexes, a cubicalcomplex can be defined either abstractly (as a poset) or geometrically (as a cellcomplex).

Definition 2.9.1. An abstract cubical complex is a finite poset (C,⊂) contain-ing an initial element ∅ and satisfying the following two conditions:

(a) for every element G ∈ C the segment [∅, G] is isomorphic to the face posetof a cube;

(b) for every two elements G1, G2 ∈ C there is a unique meet (greatest lowerbound).

Elements G ∈ C are faces of the cubical complex. If [∅, G] is the face poset ofthe k-cube Ik, then the face G is of k-dimensional. The dimension of C is themaximal dimension of its faces. The meet of any two faces G1, G2 is also calledtheir intersection and denoted G1 ∩G2.

A d-dimensional topological cube is a d-ball with a face structure defined by ahomeomorphism with the standard cube Id. A face of a topological d-cube is thusthe homeomorphic image of a face of Id.

Definition 2.9.2. A topological cubical complex is a set U of topological cubesof arbitrary dimensions which are all embedded in the same space Rn and satisfythe following conditions:


(a) every face of a cube in U belongs to U ;(b) The intersection of any two cubes in U is a face of each.

Every abstract cubical complex C has a geometric realisation, a topologicalcubical complex U whose faces form a poset isomorphic to C. Such U can beconstructed by taking a disjoint union of topological cubes corresponding to allsegments [∅, G] ⊂ C and identifying faces according to the poset relation.

From now on we shall not distinguish between abstract cubical complexes andtheir geometric realisations.

By analogy with simplicial complexes, define the f -vector of a cubical complexC by f (C) = (f0, f1, . . .) where fi is the number of i-dimensional faces. Thereare also notions of h- and g-vectors, and cubical analogues of the UBC, LBC andg-conjecture. See [2], [13] and [294] for more details and references.

The difference between Definition 2.9.2 of a geometric cubical complex andDefinition 2.2.1 of a geometric simplicial complex is that we realise abstract cubesby topological complexes rather than polytopes. This difference is substantial: ifwe replace topological cubes by combinatorial ones (i.e. by convex polytopes com-binatorially equivalent to a cube) in Definition 2.9.2, then we obtain the definitionof a polyhedral cubical complex . Although this notion is also important in combina-torial geometry, not every abstract cubical complex can be realised by a polyhedralcomplex, as shown by the next example.

Example 2.9.3. Consider the decomposition of a Mobius strip into 3 squaresshown in Fig. 2.10.

A F E B

B C D A

Figure 2.10. Cubical complex which does not admit a polyhedral realisation.

Proposition 2.9.4. The topological cubical complex shown in Fig. 2.10 doesnot admit a polyhedral cubical realisation.

Proof. Assume that such a realisation exists. Then since ABED is a convex4-gon, the points A andD are in the same halfplane with respect to the line BE, andtherefore A and D are in the same halfspace defined by the plane BCE. Similarly,since ABCF is a convex 4-gon, the points A and F are in the same halfspace withrespect to BCE. Hence, D and F are also in the same halfspace with respect toBCE. On the other hand, since CDEF is a convex 4-gon, the points D and Fmust be in different subspaces with respect to BCE. A contradiction.

The example above shows that, unlike the case of simplicial complexes, thetheory of abstract cubical complexes cannot be described by using only convex-geometric considerations. Another simple manifestation of this is the fact that not


every cubical complex may be realised as a subcomplex in a standard cube, in con-trast to simplicial complexes which are always embeddable in a standard simplex.The boundary of a triangle is the simplest example of a cubical complex not em-beddable in a cube. It is also not embeddable into the standard cubical lattice inRn (for any n). On the other hand, every cubical complex admits a cubical sub-division which is embeddable in a standard cube, as shown in the next subsection.Without subdivision the question of embeddability in a standard cube or cubicallattice is nontrivial. The importance of studying cubical maps (in particular, cu-bical embeddings) of 2-dimensional cubical complexes to the cubical lattice in R3

was pointed out by S. Novikov in connection with the 3-dimensional Ising model.In [97] necessary and sufficient conditions were obtained for a cubical complex toadmit a cubical map to the standard lattice.

Cubical subdivisions of simple polytopes and simplicial complexes.

The particular constructions of cubical complexes given here will be importantin the definition of moment-angle complexes. Neither of these constructions isparticularly new, but they are probably not well recorded in the literature (seehowever the references at the end of the section).

Any face of Im has the form

CJ⊂I = (y1, . . . , ym) ∈ Im : yj = 0 for j ∈ J, yj = 1 for j /∈ Iwhere J ⊂ I is a pair of embedded (possibly empty) subsets of [m]. We also set

(2.10) CI = C∅⊂I = (y1, . . . , ym) ∈ Im : yj = 1 for j /∈ Ito simplify the notation.

Construction 2.9.5 (canonical triangulation of Im). Denote by ∆ = ∆m−1

the simplex on [m]. We assign to a subset I = i1, . . . , ik ⊂ [m] the vertexvI = CI⊂I of Im. That is, vI = (ε1, . . . , εm) where εi = 0 if i ∈ I and εi = 1otherwise. Regarding each I as a vertex of the barycentric subdivision of ∆, we canextend the correspondence I 7→ vI to a piecewise linear embedding ic : ∆

′ → Im.Under this embedding the vertices of ∆ are mapped to the vertices of Im withexactly one zero coordinate, and the barycentre of ∆ is mapped to (0, . . . , 0) ∈ Im

(see Fig. 2.11). The image ic(∆′) is the union of m facets of Im meeting at the

vertex (0, . . . , 0). For each pair I ⊂ J , all simplices of ∆′ of the form I = I1 ⊂I2 ⊂ · · · ⊂ Ik = J are mapped to the same face CI⊂J of Im. The map ic : ∆

′ → Im

extends to cone(∆′) by mapping the cone vertex to (1, . . . , 1) ∈ Im. The image ofthe resulting map cone(ic) is the whole cube Im. Thus, cone(ic) : cone(∆′) → Im

is a PL homeomorphism which is linear on simplices of cone(∆′). This defines acanonical triangulation of Im, the ‘triangulation along the main diagonal’.

The subdivisions which appear above can be summarised as follows:

Proposition 2.9.6. The PL map cone(ic) : cone(∆′)→ Im gives rise to

(a) a cubical subdivision of ∆m−1 isomorphic to “half of the boundary of Im”(the union of facets of Im containing the zero vertex);

(b) a cubical subdivision of cone∆m−1 (which is ∆m) isomorphic to Im;(c) a simplicial subdivision of Im isomorphic to cone((∆m−1)′).

Construction 2.9.7 (cubical subdivision of a simple polytope). Let P be asimple n-polytope with m facets F1, . . . , Fm. We shall construct a piecewise linear


vertex ofcone

ic(1)

ic(2)

ic(3)ic(23)

ic(13)

ic(12)

ic(123)

12

3

12

23 13

123

ic

Figure 2.11. Taking cone over the barycentric subdivision of sim-plex defines a triangulation of the cube.

embedding of P into the standard cube Im, thereby inducing a cubical subdivisionC(P ) of P by the preimages of faces of Im.

Denote by S the set of barycentres of faces of P , including the vertices andthe barycentre of the whole polytope. This will be the vertex set of C(P ). Every(n − k)-face G of P is an intersection of k facets: G = Fi1 ∩ · · · ∩ Fik . We mapthe barycentre of G to the vertex (ε1, . . . , εm) ∈ Im, where εi = 0 if i ∈ i1, . . . , ikand εi = 1 otherwise. The resulting map S → Im can be extended linearly on thesimplices of the barycentric subdivision of P to an embedding cP : P → Im. Thecase n = 2, m = 3 is shown in Fig. 2.12.


0

A F E

G

B D

CP

cP

A F

BG

C D

E

Im

Figure 2.12. Embedding cP : P → Im for n = 2, m = 3.

The image cP (P ) ⊂ Im is the union of all faces CJ⊂I such that⋂i∈I Fi 6= ∅.

For such CJ⊂I , the preimage c−1P (CJ⊂I) is declared to be a face of the cubical com-

plex C(P ). The vertex set of c−1P (CJ⊂I) is the subset of S consisting of barycentres

of all faces between the faces G and H of P , where G =⋂j∈J Fj and H =

⋂i∈I Fi.

Therefore, faces of C(P ) correspond to pairs of embedded faces G ⊃ H of P , andwe denote them by CG⊃H . In particular, maximal (n-dimensional) faces of C(P )correspond to pairs G = P , H = v, where v is a vertex of P . For these maximalfaces we use the abbreviated notation Cv = CP⊃v.

For every vertex v = Fi1 ∩ · · · ∩ Fin ∈ P with Iv = i1, . . . , in we have

(2.11) cP (Cv) = CIv =(y1, . . . , ym) ∈ Im : yj = 1 whenever v /∈ Fj

.

We therefore obtain:

Proposition 2.9.8. A simple polytope P with m facets admits a cubical decom-position whose maximal faces Cv correspond to the vertices v ∈ P . The resultingcubical complex C(P ) embeds canonically into Im, as described by (2.11).

Lemma 2.9.9. The number of k-faces of the cubical complex C(P ) is given by

fk(C(P )

)=

n−k∑

i=0

(n− ik

)fi(P ), for 0 6 k 6 n.

Proof. The formula follows from the fact that the k-faces of C(P ) are in one-to-one correspondence with the pairs Gi+k ⊃ Hi of faces of P .

Construction 2.9.10 (cubical subdivision of a simplicial complex). Let K bea simplicial complex on [m]. Then K is naturally a subcomplex of ∆m−1 and itsbarycentric subdivision K′ is a subcomplex of (∆m−1)′. Restricting the PL mapfrom Construction 2.9.5 to K′, we obtain the embedding ic|K′ : |K′| → Im. Its imageis a cubical subcomplex in Im, which we denote cub(K). Then cub(K) is the unionof faces CI⊂J ⊂ Im over all pairs I ⊂ J of nonempty simplices of K:

(2.12) cub(K) =⋃

∅ 6=I⊂J∈K

CI⊂J ⊂ Im.


Construction 2.9.11. Since cone(K′) is a subcomplex of cone((∆m−1)′), Con-struction 2.9.5 also provides a PL embedding

cone(ic)|cone(K′) : | cone(K′)| → Im.

The image of this embedding is an n-dimensional cubical subcomplex of Im, whichwe denote cc(K). It is easy to see that

(2.13) cc(K) =⋃

I⊂J∈K

CI⊂J =⋃

J∈K

CJ .

Remark. If i ∈ [m] is not a vertex of K (a ghost vertex), then cc(K) is con-tained in the facet yi = 1 of Im.

Here is a summary of the two previous constructions.

Proposition 2.9.12. For any simplicial complex K on the set [m], there isa PL embedding of |K| into Im linear on the simplices of K′. The image of thisembedding is the cubical subcomplex (2.12). Moreover, there is a PL embedding of| coneK| into Im linear on the simplices of cone(K′), whose image is the cubicalsubcomplex (2.13).

A cubical complex C′ is called a cubical subdivision of a cubical complex C ifeach face of C′ is contained in a face of C, and each face of C is a union of finitelymany faces of C′.

Proposition 2.9.13. For every cubical complex C with q vertices, there existsa cubical subdivision that is embeddable as a subcomplex in Iq.

Proof. We first construct a simplicial complex KC which subdivides the cu-bical complex C and has the same vertices. This can be done by induction on theskeleta of C, by extending the triangulation from the k-dimensional skeleton to theinteriors of (k + 1)-dimensional faces using a generic convex function f : Rk → R,see [325, §5.1] (note that the 1-skeleton of C is already a simplicial complex). Thenapplying Construction 2.9.10 to KC we get cubical complex cub(KC) that subdividesKC and therefore C. It is embeddable into Iq by Proposition 2.9.12.

0

(a) K = 3 points

0

(b) K = ∂∆2

Figure 2.13. Cubical complex cub(K).

Example 2.9.14. The cubical complexes cub(K) and cc(K) when K is a disjointunion of 3 vertices or the boundary of a triangle are shown in Figs. 2.13–2.14.


0

(a) K = 3 points

0

(b) K = ∂∆2

Figure 2.14. Cubical complex cc(K).

Remark. Let P be a simple n-polytope, and let KP be its nerve complex.Then cc(KP ) = cP (P ), i.e. cc(KP ) coincides with the cubical complex C(P ) fromConstruction 2.9.7.

Different versions of Construction 2.9.11 can be found in [13, p. 299]. A similarconstruction was also considered in [90, p. 434]. Finally, a version of cubical sub-complex cub(K) ⊂ Im appeared in [97] in connection with the problems describedin the end of the previous subsection.

Exercises.

2.9.15. Show that the triangulation of Im from Construction 2.9.5 coincideswith the triangulation of the product of m one-dimensional simplices from Con-struction 2.2.10.


f

CHAPTER 3

Combinatorial algebra of face rings

In this chapter we collect the wealth of algebraic notions and constructionsrelated to face rings. Our choice of material and notation was guided by the topo-logical applications in the later chapters of the book. (This explains the unusual foralgebraists even grading in the polynomial rings and their homogeneous quotients,and also the nonpositive homological grading in free resolutions and Tor.) In thefirst sections we review standard results and constructions of combinatorial commu-tative algebra, including the Tor-algebras and algebraic Betti numbers of face rings,Cohen–Macaulay and Gorenstein complexes. The later sections contain some morerecent developments, including the face rings of simplicial posets, different charac-terisations of Cohen–Macaulay and Gorenstein simplicial posets in terms of theirface rings and h-vectors, and generalisations of the Dehn–Sommerville relations.Although all these algebraic and combinatorial results have a strong topologicalflavour and were indeed originally motivated by topological constructions, we havetried to keep this chapter mostly algebraic and do not require much topologicalknowledge from the reader here.

The preliminary algebraic material of a more general sort, not directly or ex-clusively related to the face rings (such as resolutions and the functor Tor, andCohen–Macaulay rings) is collected in Appendix A.

Alongside the monograph by Stanley [293], an extensive survey of Cohen–Macaulay rings by Bruns and Herzog [45] and a more recent monograph [225] byMiller and Sturmfels may be recommended for a deeper study of algebraic methodsin combinatorics.

We use the common notation k for the ground ring, which is always assumedto be the ring Z of integers or a field. The former is preferable for topologicalapplications, but the latter is more common in the algebraic literature. We shalloften refer to k-modules as ‘k-vector spaces’; in the case k = Z the latter means‘abelian groups’.

We assume graded commutativity instead of commutativity; algebras commuta-tive in the standard sense will be those whose nontrivial graded components appearonly in even degrees. In particular, the polynomial algebra k[v1, . . . , vm], which weoften abbreviate to k[m], has deg vi = 2. The exterior algebra Λ[u1, . . . , um] hasdeg ui = 1. Given a subset I = i1, . . . , ik ⊂ [m] we denote by vI the square-freemonomial vi1 · · · vik in k[m]. We also denote by uI the exterior monomial ui1 · · ·uikwhere i1 < · · · < ik.

97

98 3. COMBINATORIAL ALGEBRA OF FACE RINGS

3.1. Face rings of simplicial complexes

Definition 3.1.1. The face ring (or the Stanley–Reisner ring) of a simplicialcomplex K on the set [m] is the quotient graded ring

k[K] = k[v1, . . . , vm]/IK,where IK = (vI : I /∈ K) is the ideal generated by those monomials vI for which Iis not a simplex of K. The ideal IK is known as the Stanley–Reisner ideal of K.

Example 3.1.2.1. Let K be the 2-dimensional simplicial complex shown in Fig. 3.1. Then

IK = (v1v5, v3v4, v1v2v3, v2v4v5).

rr rr

r1 3

2

4 5

Figure 3.1.

2. The face ring k[K] is a quadratic algebra (that is, the ideal IK is generatedby quadratic monomials) if and only if K is a flag complex (an exercise).

3. Let K1 ∗ K2 be the join of K1 and K2 (see Construction 2.2.8). Then

k[K1 ∗ K2] = k[K1]⊗ k[K2].

Here and below ⊗ denotes the tensor product over k.

We note that IK is a monomial ideal, and it has a basis consisting of square-freemonomials vI corresponding to the missing faces of K.

Proposition 3.1.3. Every square-free monomial ideal I in the polynomial ringis the Stanley–Reisner ideal of a simplicial complex K.

Proof. We set

K = I ⊂ [m] : vI /∈ I.Then K is a simplicial complex and I = IK.

Let P be a simple n-polytope and let KP be its nerve complex (see Exam-ple 2.2.4). We define the face ring k[P ] as the face ring of KP . Explicitly,

k[P ] = k[v1, . . . , vm]/IP ,where IP is the ideal generated by those square-free monomials vi1vi2 · · · vis whosecorresponding facets intersect trivially, Fi1 ∩ · · · ∩ Fis = ∅.

3.1. FACE RINGS OF SIMPLICIAL COMPLEXES 99

Example 3.1.4.1. Let P be an n-simplex (viewed as a simple polytope). Then

k[P ] = k[v1, . . . , vn+1]/(v1v2 · · · vn+1).

2. Let P be a 3-cube I3. Then

k[P ] = k[v1, v2, . . . , v6]/(v1v4, v2v5, v3v6).

3. Let P be an m-gon, m > 4. Then

IP = (vivj : i− j 6= 0,±1 mod m).

4. Given two simple polytopes P1 and P2, we have

k[P1 × P2] = k[P1]⊗ k[P2].

Proposition 3.1.5. Let ϕ : K → L be a simplicial map between simplicialcomplexes K and L on the vertex sets [m] and [l] respectively. Define the mapϕ∗ : k[w1, . . . , wl]→ k[v1, . . . , vm] by

ϕ∗(wj) =∑

i∈ϕ−1(j)

vi.

Then ϕ∗ descends to a homomorphism k[L] → k[K], which we continue to de-note ϕ∗.

Proof. We only need to check that ϕ∗(IL) ⊂ IK. Suppose J = j1, . . . , js ⊂[l] is not a simplex of L. We have

ϕ∗(wj1 . . . wjs) =∑

i1∈ϕ−1(j1),...,is∈ϕ−1(js)

vi1 . . . vis .

We claim that the right hand side above belongs to IK, i.e. for any monomialvi1 . . . vis in the right hand side the set I = i1, . . . , is is not a simplex of K.Indeed, otherwise we would have ϕ(I) = J ∈ L by the definition of a simplicialmap, which contradicts the assumption.

Example 3.1.6. The face ring of the barycentric subdivision K′ of K is

k[K′] = k[bI : I ∈ K \∅]/IK′ ,

where bI is the polynomial generator of degree 2 corresponding to a nonemptysimplex I ∈ K, and IK′ is generated by quadratic monomials bIbJ for which I 6⊂ Jand J 6⊂ I. The simplicial map ∇ : K′ → K from Example 2.3.3 induces a map ∇∗

of the face ring, given on the generators vj ∈ k[K] by

∇∗(vj) =∑

I∈K : minI=j

bI .

Example 3.1.7. The nondegenerate map K′ → ∆n−1 from Example 2.3.2induces the following map of the corresponding face rings:

k[v1, . . . , vn] −→ k[K′]

vi 7−→∑

|I|=i

bI .

This defines a canonical k[v1, . . . , vn]-module structure on k[K′].


An important tool arising from the functoriality of the face ring is the restrictionhomomorphism. For any simplex I ∈ K, the corresponding full subcomplex KI is∆|I|−1 and k[KI ] is the polynomial ring k[vi : i ∈ I] on |I| generators. The inclusionKI ⊂ K induces the restriction homomorphism

sI : k[K]→ k[vi : i ∈ I],which maps vi to zero whenever i /∈ I.

The following simple proposition will be used in several algebraic and topolo-gical arguments of the later chapters.

Proposition 3.1.8. The direct sum

s =⊕I∈K

sI : k[K] −→⊕

I∈K

k[vi : i ∈ I]

of all restriction maps is a monomorphism.

Proof. Consider the composite map

k[v1, . . . , vm]p−−−−→ k[K] s−−−−→ ⊕

I∈K k[vi : i ∈ I]where p is the quotient projection. Suppose s p(Q) = 0 where Q = Q(v1, . . . , vm)is a polynomial. Then for any monomial vα1

i1· · · vαk

ikwhich enters Q with a nonzero

coefficient we have I = i1, . . . , ik /∈ K (as otherwise the Ith component of theimage under s p is nonzero). Hence p(Q) = 0 and s is injective.

Proposition 3.1.9. The face ring k[K] has the k-vector space basis consistingof monomials vα1

j1· · · vαk

jkwhere αi > 0 and j1, . . . , jk ∈ K.

Proof. Indeed, the polynomial algebra k[m] has the k-vector space basis con-sisting of all monomials vα1

j1· · · vαk

jk, and such a monomial maps to zero under the

projection k[m]→ k[K] precisely when j1, . . . , jk /∈ K.

Recall that the Poincare series of a nonnegatively graded k-vector space V =⊕∞i=0 V

i is given by F (V ;λ) =∑∞

i=0(dimk Vi)λi. Since k[K] is graded by even

integers, its Poincare series is even.

Theorem 3.1.10 (Stanley). Let K be an (n−1)-dimensional simplicial complexwith f -vector (f0, . . . , fn−1) and h-vector (h0, . . . , hn). Then the Poincare series ofthe face ring k[K] is

F(k[K];λ

)=

n∑

k=0

fk−1

(λ2

1− λ2)k

=h0 + h1λ

2 + · · ·+ hnλ2n

(1− λ2)n .

Proof. By Proposition 3.1.9, a (k − 1)-dimensional simplex i1, . . . , ik ∈ Kcontributes a summand λ2k

(1−λ2)k to the Poincare series of K (this summand is just

the Poincare series of the subspace generated by monomials vα1

i1· · · vαk

ikwith positive

exponents αi). This proves the first identity, and the second follows from (2.3).

Example 3.1.11.1. Let K = ∆n−1. Then fi =

(ni+1

)for −1 6 i 6 n − 1, h0 = 1 and hi = 0

for i > 0. Since every subset of [n] is a simplex of ∆n−1, we have k[∆n−1] =k[v1, . . . , vn] and F (k[∆n−1];λ) = (1− λ2)−n.

3.1. FACE RINGS OF SIMPLICIAL COMPLEXES 101

2. Let K = ∂∆n be the boundary of an n-simplex. Then hi = 1 for 0 6 i 6 n,and k[∂∆n] = k[v1, . . . , vn+1]/(v1v2 · · · vn+1). By Theorem 3.1.10,

F(k[∂∆n];λ

)=

1 + λ2 + · · ·+ λ2n

(1− λ2)n .

The affine algebraic variety corresponding to the commutative finitely generatedk-algebra k[K] = k[m]/IK (i.e. the set of common zeros of elements of IK, viewedas algebraic functions on k

m) can be easily identified as follows.

Proposition 3.1.12. The affine variety corresponding to k[K] is given by

X(K) =⋃

I∈K

SI ,

where SI = k〈ei : i ∈ I〉 is the coordinate subspace in km spanned by the set of

standard basis vectors corresponding to I.

Proof. The statement obviously holds in the case K = ∆m−1. So we assume

K 6= ∆m−1. We shall use the following notation from Section 2.4: I = [m] \ I, the

complement of I ⊂ [m], and K = I ∈ [m] : I /∈ K, the dual complex of K. Givena point z = (z1, . . . , zm) ∈ k

m, we denote by

ω(z ) = i : zi = 0 ⊂ [m],

the set of zero coordinates of z .By the definition of the algebraic variety X(K) corresponding to k[K],

X(K) =⋂

J /∈K

⋃

j∈J

z : zj = 0 =⋂

J /∈K

z : ω(z ) ∩ J 6= ∅

=⋂

J∈K

z : ω(z ) 6⊂ J

=z : ω(z ) /∈ K

.

On the other hand,

⋃

I∈K

SI =⋃

I∈K

⋂

j∈I

z : zj = 0 =⋃

I∈K

z : I ⊂ ω(z )

=⋃

I /∈K

z : ω(z ) ⊃ I

=z : ω(z ) /∈ K

.

The required identity follows by comparing the two formulae above.

Remark. The variety X(K) is an example of an arrangement of coordinatesubspaces, which will be studied further in Section 4.7.

We finish this section with a result showing that the face ring determines itsunderlying simplicial complex:

Theorem 3.1.13 (Bruns–Gubeladze [44]). Let k be a field, and K1 and K2

be two simplicial complexes on the vertex sets [m1] and [m2] respectively. Supposek[K1] and k[K2] are isomorphic as k-algebras. Then there exists a bijective map[m1]→ [m2] which induces an isomorphism between K1 and K2.


Proof. Let f : k[K1] → k[K2] be an isomorphism of k-algebras. An easyargument shows that we can assume that f is a graded isomorphism (an exercise,or see [44, p. 316]).

Since f is graded, by restriction to the linear components we observe thatm1 = m2 and that f is induced by a linear isomorphism F : k[m1] → k[m2]. Thisis described by the commutative diagram

k[v1, . . . , vm1 ]F //

k[v1, . . . , vm2 ]

k[K1]

f // k[K2]

By passing to the associated affine varieties, we observe that the isomorphismf∗ : X(K2)→ X(K1) is the restriction of the k-linear isomorphism F ∗ : km2 → k

m1 .This is described by the commutative diagram

km2

F∗

// km1

X(K2)

OO

f∗

// X(K1)

OO

The isomorphism f∗ establishes a bijective correspondence

Φ : maximal faces of K2 → maximal faces of K1which is defined by the formula f∗(SI) = SΦ(I), where I is a maximal face of K2.It is also clear that |Φ(I)| = |I| = dimSI .

We denote by P1 the intersection poset of the subspaces SI , I ∈ K1, withrespect to inclusion (i.e. the elements of P1 are nonempty intersections SI1∩· · ·∩SIkwith Ij ∈ K1). The poset P1 can be also viewed as the intersection poset of themaximal faces of K1. We define the poset P2 corresponding to K2 similarly. ThecorrespondenceΦ obviously extends to an isomorphism of posets Φ : P2 → P1, whichpreserves the dimension of spaces (or the number of elements in the intersectionsof maximal faces).

Now introduce the following equivalence relation on the vertex sets [m1] and[m2]: for i1, i2 ∈ [m1] (or j1, j2 ∈ [m2]) we put i1 ∼ i2 if and only if the two setsof maximal faces K1 containing i1 and i2 respectively coincide (and similarly for j1and j2). The equivalence classes in [m1] are the minimal (with respect to inclusion)nonempty intersections of maximal faces of K1, and similarly for [m2]. Since Φis an isomorphism of posets, the two systems of equivalence classes are in naturalbijective correspondence, and the corresponding equivalence classes have the samenumbers of elements. This gives rise to the bijective map ϕ : [m2] → [m1] whichsatisfies the condition that i ∈ I if and only if ϕ(i) ∈ Φ(I), where i ∈ [m2] andI ∈ K2 is a maximal face. Since any face of a simplicial complex is contained in amaximal face, we obtain that ψ = ϕ−1 : [m1]→ [m2] is the required map.

Exercises.

3.1.14. Show that the Stanley–Reisner ideal IK is generated by quadratic mono-mials if and only if K is a flag complex.

3.2. TOR-ALGEBRAS AND BETTI NUMBERS 103

3.1.15 (see [261, (4.7)]). Let cat(K) be the face category of K (objects aresimplices, morphisms are inclusions), catop(K) the opposite category (in which themorphisms are reverted), and cga the category of commutative graded algebras.(See Appendix C.1 for basics of categories and diagrams.) Consider the diagram

k[ · ]K : catop(K) −→ cga,

I 7−→ k[vi : i ∈ I]whose value on a morphism I ⊂ J is the surjection k[vj : j ∈ J ] → k[vi : i ∈ I]sending each vj with j /∈ I to zero. Show that

k[K] = limk[ · ]K

where the limit is taken in the category cga.

3.1.16. If k[K1] and k[K2] are isomorphic as k-algebras, then there is alsoa graded isomorphism k[K1] → k[K2]. (Hint: Show first that k[K1] and k[K2]are isomorphic as augmented k-algebras, and then pass to the associated gradedalgebras with respect to the augmentation ideals.)

3.2. Tor-algebras and Betti numbers

The algebraic Betti numbers of the face ring k[K] are the dimensions of theTor-groups of k[K] viewed as a module over the polynomial ring. These basichomological invariants of a simplicial complex K appear to be of great importanceboth for combinatorial commutative algebra and toric topology.

The face ring k[K] acquires a canonical k[m]-module structure via the quotientprojection k[m]→ k[K]. We therefore may consider the corresponding Tor-modules(see Appendix, Section A.2):

Tork[v1,...,vm]

(k[K],k

)=⊕

i,j>0

Tor−i,2jk[v1,...,vm]

(k[K],k

).

From Lemma A.2.10 we obtain that Tork[m](k[K],k) is a bigraded algebra in anatural way, and there is the following isomorphism of bigraded algebras:

Tork[v1,...,vm]

(k[K],k

) ∼= H[Λ[u1, . . . , um]⊗ k[K], d

],

where the bigrading and differential on the right hand side are given by

(3.1)bideg ui = (−1, 2), bideg vi = (0, 2),

dui = vi, dvi = 0.

Definition 3.2.1. We refer to Tork[v1,...,vm](k[K],k) as the Tor-algebra of asimplicial complex K.

The bigraded Betti numbers of k[K] are defined by

(3.2) β−i,2j(k[K]

)= dimk Tor

−i,2jk[v1,...,vm]

(k[K],k

), for i, j > 0.

We also set

β−i(k[K]

)= dimk Tor

−ik[v1,...,vm]

(k[K],k

)=∑

j

β−i,2j(k[K]

).

The Tor-algebra has the following functorial property:


Proposition 3.2.2. A simplicial map ϕ : K → L between simplicial complexeson the sets [m] and [l] respectively induces a homomorphism

ϕ∗Tor : Tork[w1,...,wl]

(k[L],k

)→ Tork[v1,...,vm]

(k[K],k

)

of the corresponding Tor-algebras.

Proof. This follows from Proposition 3.1.5 and Theorem A.2.5 (b).

Consider the minimal resolution (Rmin, d) of the k[m]-module k[K] (see Con-struction A.2.2). Then R0

min∼= 1 · k[m] is a free module with one generator

of degree 0. The basis of R−1min is a minimal generator set for IK, and these

minimal generators correspond to the missing faces of K. Given a missing facei1, . . . , ik ⊂ [m], denote by ri1,...,ik the corresponding generator of R−1

min. Then the

map d : R−1min → R0

min takes ri1,...,ik to vi1 . . . vik . By Proposition A.2.6, β−1,2j(k[K])is equal to the number of missing faces with j elements.

Example 3.2.3. Let K = rr rr1 2

4 3

, the boundary of a 4-gon. Then

k[K] ∼= k[v1, . . . , v4]/(v1v3, v2v4).

Let us construct a minimal resolution of k[K]. The module R0min has one generator

1 (of degree 0). The module R−1min has two generators r13 and r24 of degree 4,

and the differential d : R−1min → R0

min takes r13 to v1v3 and r24 to v2v4. The kernel

R−1min → R0

min is generated by one element v2v4r13 − v1v3r24. Hence, R−2min has

one generator of degree 8, which we denote by a, and the map d : R−2min → R−1

min isinjective and takes a to v2v4r13 − v1v3r24. Thus, the minimal resolution is

0 −−−−→ R−2min −−−−→ R−1

min −−−−→ R0min −−−−→ M −−−−→ 0,

where rankR0min = β0,0(k[K]) = 1, rankR−1

min = β−1,4(k[K]) = 2, rankR−2min =

β−2,8(k[K]) = 1.

The following fundamental result of Hochster reduces the calculation of theBetti numbers β−i,2j(k[K]) to the calculation of reduced simplicial cohomology offull subcomplexes in K.

Theorem 3.2.4 (Hochster [162]). We have

Tor−i,2jk[v1,...,vm]

(k[K],k

)=

⊕

J⊂[m] : |J|=j

Hj−i−1(KJ ;k),

where KJ is the full subcomplex of K obtained by restricting to J ⊂ [m]. We assume

H−1(K∅;k) = k above.

We shall give a proof of Hochster’s formula following [257]. The idea is to firstreduce the Koszul algebra (Λ[u1, . . . , um]⊗ k[K], d) to a certain finite dimensionalquotient R∗(K), without changing the cohomology, and then identify R∗(K) withthe sum of simplicial cochain complexes of all full subcomplexes in K. The alge-bra R∗(K) will also be used in the cohomological calculations for moment-anglecomplexes in Chapter 4.

We use simplified notation uJvI for a monomial uJ ⊗ vI in the Koszul algebraΛ[u1, . . . , um]⊗ k[K].


Construction 3.2.5. We introduce the quotient algebra

R∗(K) = Λ[u1, . . . , um]⊗ k[K]/(v2i = uivi = 0, 1 6 i 6 m).

Since the ideal generated by v2i and uivi is homogeneous and invariant with respectto the differential (since d(uivi) = 0 and d(v2i ) = 0), we obtain that R∗(K) hasdifferential and bigrading (3.1). We also have the quotient projection

: Λ[u1, . . . , um]⊗ k[K]→ R∗(K).By definition, the algebra R∗(K) has a k-vector space basis consisting of monomialsuJvI where J ⊂ [m], I ∈ K and J ∩ I = ∅. Therefore,

(3.3) dimkR−p,2q = fq−p−1

(m−q+p

p

),

where (f0, f1, . . . , fn−1) is the f -vector of K and f−1 = 1. We have a k-linear map

ι : R∗(K)→ Λ[u1, . . . , um]⊗ k[K]sending each uJvI identically. The map ι commutes with the differentials, andtherefore defines a homomorphism of bigraded differential k-vector spaces satisfyingthe relation · ι = id. Note that ι is not a map of algebras.

Lemma 3.2.6. The projection homomorphism : Λ[u1, . . . , um]⊗k[K]→ R∗(K)induces an isomorphism in cohomology.

Proof. The argument is similar to that used for the Koszul resolution (seeConstruction A.2.4). We shall construct a cochain homotopy between the maps idand ι · from Λ[u1, . . . , um]⊗ k[K] to itself, that is, a map s satisfying the identity

(3.4) ds+ sd = id− ι · .We first consider the case K = ∆m−1. Then Λ[u1, . . . , um] ⊗ k[∆m−1] is the

Koszul resolution (A.5), which will be denoted by

(3.5) E = Em = Λ[u1, . . . , um]⊗ k[v1, . . . , vm],

and the algebra R∗(∆m−1) is isomorphic to

(3.6)(Λ[u]⊗ k[v]

/(v2, uv)

)⊗m.

For m = 1, we define the map s1 : E0,∗1 = k[v]→ E−1,∗

1 by the formula

s1(a0 + a1v + · · ·+ ajvj) = u(a2v + a3v

2 + · · ·+ ajvj−1).

We need to check identity (3.4) for x = a0 + a1v + · · · + ajvj ∈ E0,∗

1 and for

ux ∈ E−1,∗1 , as each element of E1 is the sum of elements of these two types. In the

first case we have ds1x = x− a0− a1v = x− ιx, and s1dx = 0. In the second case,i.e. for ux ∈ E−1,∗

1 , we have ds1(ux) = 0, and s1d(ux) = ux − a0u = ux− ι(ux).In both cases (3.4) holds.

Now we may assume by induction that a cochain homotopy sm : Em → Em hasbeen already constructed for m = k − 1. Since Ek = Ek−1 ⊗ E1, k = k−1 ⊗ 1and ιk = ιk−1 ⊗ ι1, a direct calculation shows that the map

(3.7) sk = sk−1 ⊗ id + ιk−1k−1 ⊗ s1is a cochain homotopy between id and ιkk, which finishes the proof for K = ∆m−1.

In the case of arbitrary K the algebras Λ[u1, . . . , um]⊗k[K] and R∗(K) are ob-tained by factorising (3.5) and (3.6) respectively by the ideal IK in Λ[u1, . . . , um]⊗k[m]. Observe that IK is generated by vI with I /∈ K as an ideal, and it has


a k-vector space basis of monomials uJvα1

i1· · · vαk

ikwith I = i1, . . . , ik /∈ K and

αi > 0. We need to check that

d(IK) ⊂ IK, ι(IK) ⊂ IK, s(IK) ⊂ IK.The first inclusion is obvious: since d is a derivation, we only need to check thatdvI ∈ IK for I /∈ K, but dvI = 0. The second inclusion is also clear, since

ι(uJvα1

i1· · · vαk

ik) =

uJvi1 · · · vik , if αi = 1 and J ∩ i1, . . . , ik = ∅;

0, otherwise.

It remains to check the third inclusion. By expanding the inductive formula (3.7)we obtain

sm = s1 ⊗ id⊗ · · · ⊗ id + ι11 ⊗ s1 ⊗ id⊗ · · · ⊗ id + ι11 ⊗ · · · ⊗ ι11 ⊗ s1.It follows that

sm(uJvα1

i1· · · vαk

ik) =

∑

p : αp>1

±uJuipvα1

i1· · · vαp−1

ip· · · vαk

ik.

Therefore, s(IK) ⊂ IK, and identity (3.4) holds in Λ[u1, . . . , um]⊗ k[K].

As an immediate consequence of Lemma 3.2.6 we obtain

Corollary 3.2.7. We have that β−i,2j(k[K]

)= 0 if i > m or j > m.

Proof. Indeed, R−i,2j(K) = 0 if either i or j is greater than m.

Now, in order to prove Theorem 3.2.4, we need to show that the cohomology ofR∗(K) is isomorphic to the direct sum of the reduced cohomology of the full subcom-plexes on the right hand side of Hochster’s formula. We shall see that this is trueeven without passing to cohomology, i.e. R∗(K) is isomorphic to

⊕I⊂m C

∗(KI),with the appropriate shift in dimensions, where C∗ denotes the simplicial cochaingroups. To do this, it is convenient to refine the grading in k[K] as follows.

Construction 3.2.8 (multigraded structure in face rings and Tor-algebras). Amultigrading (more precisely, an Nm-grading) is defined in k[v1, . . . , vm] by setting

mdeg vi11 · · · vimm = (2i1, . . . , 2im).

Since k[K] is the quotient of the polynomial ring by a monomial ideal, it inheritsthe multigrading. We may assume that all free modules in the resolution (A.2)are multigraded and the differentials preserve the multidegree. Then the algebraTork[m](k[K],k) acquires the canonical Z⊕ Nm-grading, i.e.

Tork[v1,...,vm]

(k[K],k

)=

⊕

i>0, a∈Nm

Tor−i,2ak[v1,...,vm]

(k[K],k

).

The differential algebra R∗(K) also acquires a Z ⊕ Nm-grading, and Lemma 3.2.6implies that

(3.8) Tor−i,2ak[v1,...,vm]

(k[K],k

) ∼= H−i,2a (R∗(K), d).

We may view a subset J ⊂ [m] as a (0, 1)-vector in Nm whose jth coordinateis 1 if j ∈ J and is 0 otherwise. Then there is the following multigraded version ofHochster’s formula:


Theorem 3.2.9. For any subset J ⊂ [m] we have

Tor−i,2Jk[v1,...,vm]

(k[K],k

) ∼= H |J|−i−1(KJ ),

and Tor−i,2ak[m] (k[K],k) = 0 if a is not a (0, 1)-vector.

Proof of Theorem 3.2.4 and Theorem 3.2.9. Let Cq(KJ ) denote the qthsimplicial cochain group with coefficients in k. Denote by αL ∈ Cp−1(KJ ) the basiscochain corresponding to an oriented simplex L = (l1, . . . , lp) ∈ KJ ; it takes value 1on L and vanishes on all other simplices. Now we define a k-linear map

(3.9)f : Cp−1(KJ ) −→ Rp−|J|,2J(K),

αL 7−→ ε(L, J)uJ\LvL,

where ε(L, J) is the sign defined by

ε(L, J) =∏

j∈L

ε(j, J),

and ε(j, J) = (−1)r−1 if j is the rth element of the set J ⊂ [m], written in increasingorder. Obviously, f is an isomorphism of k-vector spaces, and a direct check showsthat it commutes with the differentials. Indeed, we have

f(dαL) = f( ∑

j∈J\L, j∪L∈KJ

ε(j, j ∪ L)αj∪L)

=∑

j∈J\L

ε(j ∪ L, J)ε(j, j ∪ L)uJ\(j∪L)vj∪L

(note that vj∪L ∈ k[K], and hence it is zero unless j∪L ∈ KJ ). On the other hand,

df(αL) =∑

j∈J\L

ε(L, J)ε(j, J \ L)uJ\(j∪L)vj∪L.

By the definition of ε(L, J),

ε(j ∪ L, J)ε(j, j ∪ L) = ε(L, J)ε(j, J)ε(j, j ∪ L) = ε(L, J)ε(j, J \ L),which implies that f(dαL) = df(αL). Therefore, f together with the map k →R−|J|,2J(K), 1 7→ uJ , defines an isomorphism of cochain complexes

0→ kd−→ C0(KJ) d−→ · · · d−→ Cp−1(KJ ) d−→ · · ·

↓∼= ↓∼= ↓∼=0→ R−|J|,2J(K) d−→ R1−|J|,2J(K) d−→ · · · d−→ Rp−|J|,2J(K) d−→ · · ·

Then it follows from (3.8) that

Hp−1(KJ) ∼= Torp−|J|,2Jk[v1,...,vm]

(k[K],k

),

which is equivalent to the first isomorphism of Theorem 3.2.9. Since R−i,2a (K) = 0

if a is not a (0, 1)-vector, Tor−i,2ak[m] (k[K],k) vanishes for such a .

Since Tork[m](k[K],k) is an algebra, the isomorphisms of Theorem 3.2.4 turnthe direct sum

(3.10)⊕

p>0

J⊂[m]

Hp−1(KJ )


into a (multigraded) k-algebra. Consider the product in the simplicial cochains offull subcomplexes given by

(3.11)

µ : Cp−1(KI)⊗ Cq−1(KJ ) −→ Cp+q−1(KI∪J ),

αL ⊗ αM 7−→αL⊔M , if I ∩ J = ∅;0, otherwise.

Here αL⊔M ∈ Cp+q−1(KI⊔J ) denotes the basis simplicial cochain corresponding toL ⊔M if the latter is a simplex of KI⊔J and zero otherwise. If I ∩ J = ∅, thenKI⊔J is a subcomplex in the join KI ∗ KJ , and the above product is the restrictionto KI⊔J of the standard exterior product

Cp−1(KI)⊗ Cq−1(KJ ) −→ Cp+q−1(KI ∗ KJ ).

Proposition 3.2.10. The product in the direct sum⊕

p>0, J⊂[m] Hp−1(KJ )

induced by the isomorphisms from Hochster’s theorem coincides up to a sign withthe product given by (3.11).

Proof. This is a direct calculation. We use the isomorphism f given by (3.9):

αL · αM = f−1(f(αL) · f(αM )

)= f−1

(ε(L, I)uI\LvL ε(M,J)uJ\MvM

)

If I∩J 6= ∅, then the product uI\LvLuJ\MvM is zero in R∗(K). Otherwise we havethat uI\LvLuJ\MvM = ζ u(I∪J)\(L∪M)vL∪M , where ζ =

∏k∈I\L ε(k, k∪J \M), and

we can continue the above identity as

αL · αM = ε(L, I) ε(M,J) ζ ε(L ∪M, I ∪ J)αL⊔M .

Note that this calculation also gives the explicit value for the correcting sign, butwe shall not need this.

Let P be a simple polytope. The multigraded components of Tork[m](k[P ],k)can be expressed directly in terms of P as follows. Let F1, . . . , Fm be the set offacets of P . Given I ⊂ [m], we define the following subset of the boundary of P :

PI =⋃

i∈I

Fi.

Proposition 3.2.11. For any subset J ⊂ [m] we have


(k[P ],k

) ∼= H |J|−i−1(PJ),

and Tor−i,2ak[m] (k[P ],k) = 0 if a is not a (0, 1)-vector.

Proof. Let K = KP be the nerve complex of P . Then the statement followsfrom Theorem 3.2.9 and the fact that KJ is a deformation retract of PJ . Thelatter is because P is simple, and therefore, PJ =

⋃i∈J FJ =

⋃i∈J stK′i (by

Proposition 2.3.8), which is the combinatorial neighbourhood of (KJ )′ in K′.

For a description of the multiplication in Tork[m](k[P ],k) in terms of P , seeExercise 3.2.14.


Example 3.2.12.

1. Let P be a 4-gon, so that KP = rr rr1 2

4 3

, as in Example 3.2.3. This time wecalculate the Betti numbers β−i,2j(k[P ]) using Hochster’s formula. We have

β0,0(k[P ]

)= dim H−1(∅) = 1 1

β−1,4(k[P ]

)= dim H0

(P1,3

)⊕ H0

(P2,4

)= 2 u1v3, u2v4

β−2,8(k[P ]

)= dim H1

(P1,2,3,4

)= 1 u1u2v3v4

where in the right column we include cocycles in the Koszul algebra Λ[u1, . . . , um]⊗k[P ] representing generators of the corresponding cohomology groups. All otherBetti numbers are zero. We have a nontrivial product [u1v3] · [u2v4] = [u1u2v3v4];all other products of positive-dimensional classes are zero. Note that in this exampleall Tor-groups have bases represented by monomials in the Koszul algebra. This isnot the case in general, as is shown by the next example.

2. Now let K = r r r r1 2 3 4 be the union of two segments. Then thegenerator of

Tor−3,8k[v1,...,v4]

(k[K],k

) ∼= H0(K1,2,3,4;k

) ∼= k

is represented by the cocycle u1u2u3v4 − u1u2u4v3 in the Koszul algebra, and itcannot be represented by a monomial.

3. Let us calculate the Betti numbers (both bigraded and multigraded) of k[K]for the complex shown in Fig. 3.1, using Hochster’s formula. We have

β0,0 = dim H0(∅) = 1,

β−1,4 = β−1,(2,0,0,0,2) + β−1,(0,0,2,2,0) = dim H0(K1,5

)⊕ H0

(K3,4

)= 2,

β−1,6 = β−1,(2,2,2,0,0) + β−1,(0,2,0,2,2) = dim H1(K1,2,3

)⊕ H1

(K2,4,5

)= 2,

β−2,8 = β−2,(0,2,2,2,2) + β−2,(2,0,2,2,2) + · · ·+ β−2,(2,2,2,2,0)

= dim H1(K2,3,4,5

)⊕ · · · ⊕ H1

(K1,2,3,4

)= 5,

β−3,10 = β−3,(2,2,2,2,2) = dim H1(K1,2,3,4,5

)= 2.

All other Betti numbers are zero.4. LetK be a triangulation of the real projective plane RP 2 withm vertices (the

minimal example has m = 6, see Fig. 3.2, where the vertices with the same labelsare identified, and the boundary edges are identified according to the orientationshown). Then, by Hochster’s formula,

5

5

6

4

4

6

2

3 1

❨

Figure 3.2. 6-vertex triangulation of RP 2.


Tor3−m,2mk[v1,...,vm]

(k[K],k

)= H2(K[m];k) = H2(RP 2;k) = 0

if the characteristic of k is not 2. On the other hand,

Tor3−m,2mZ2[v1,...,vm]

(Z2[K],Z2

)= H2(K[m];Z2) = H2(RP 2;Z2) = Z2.

This example shows that the Tor-groups of k[K], and even the algebraic Bettinumbers, depend on k. A similar example shows that TorZ[m](Z[K],Z) may havean arbitrary amount of additive torsion. (This is a well-known fact for the usualcohomology of spaces, and so we may take K to be a triangulation of a space withthe appropriate torsion in cohomology.)

Exercises.

3.2.13. Let P be a pentagon. Calculate the bigraded Betti numbers of k[P ]and the multiplication in Tork[m](k[P ],k)

(a) using algebra R∗(KP ) and Lemma 3.2.6;(b) using Hochster’s theorem and Proposition 3.2.10,

and compare the results.

3.2.14. Use Proposition 3.2.11 and the isomorphism

H |J|−i−1(PJ ) ∼= H |J|−i(P, PJ )

to show that the multiplication induced from Tork[m](k[P ],k) in the direct sum⊕

p>0

J⊂[m]

Hp(P, PJ )

comes from the standard exterior multiplication

Hp(P, PI)⊗Hq(P, PJ ) −→ Hp+q(P, PI ∪ PJ)when I ∩ J = ∅ and is zero otherwise.

3.2.15. Complete the details in the following algebraic proof of the Alexan-der duality (Theorem 2.4.5); this argument goes back to the original work ofHochster [162]:

1. Choose J /∈ K, that is, J = [m] \ J ∈ K, and show that for any L =l1, . . . , lq ⊂ J ,

J \ L /∈ K ⇐⇒ L ∈ lkK J .

2. Consider the Koszul algebra

S(K) =[Λ[u1, . . . , um]⊗ IK, d

]

of the Stanley–Reisner ideal IK (see Lemma A.2.10 and the remark after it), andshow that its multigraded component S−q,2J(K) has a k-basis consisting of mono-

mials uLvJ\L where L ∈ lkK J .3. Consider the k-vector space isomorphism

g : Cq−1(lkK J) −→ S−q,2J(K),[L] 7−→ uLvJ\L,

where [L] ∈ Cq−1(lkK J) is the basis simplicial chain corresponding to L. Show thatg commutes with the differentials, and therefore defines an isomorphism of chaincomplexes (in analogy with (3.9), but with no correction sign).

3.3. COHEN–MACAULAY COMPLEXES 111

4. Deduce that

Hq−1(lkK J)∼= Tor−q,2J

k[m] (IK,k) ∼= Tor−q−1,2Jk[m] (k[K],k) ∼= H |J|−q−2(KJ ),

where the first isomorphism is obtained by passing to homology in step 3, thesecond follows from the long exact sequence of Theorem A.2.5 (e), and the third isTheorem 3.2.4. It remains to note that the resulting isomorphism is equivalent tothat of Corollary 2.4.6.

3.3. Cohen–Macaulay complexes

It is usually quite difficult to determine whether a given ring is Cohen–Macaulay(see Appendix, Section A.3). One of the key results of combinatorial commu-tative algebra, the Reisner Theorem, gives an effective criterion for the Cohen–Macaulayness of face rings, in terms of simplicial cohomology of K. A reformula-tion of Reisner’s criterion, due to Munkres and Stanley, tells us that the Cohen–Macaulayness of the face ring k[K] is a topological property of K, i.e. it dependsonly on the topology of the realisation |K|. These results have many importantapplications in both combinatorial commutative algebra and toric topology.

Here we assume that k is a field, unless otherwise stated. If K is of dimensionn − 1, then the Krull dimension of k[K] is n (an exercise). We start with thefollowing combinatorial description of homogeneous systems of parameters (hsop’s)in k[K] in terms of the restriction homomorphisms sI : k[K] → k[vi : i ∈ I] (seeProposition 3.1.8).

Lemma 3.3.1. Let K be a simplicial complex of dimension n − 1. A sequenceof homogeneous elements t = (t1, . . . , tn) of the face ring k[K] is a homogeneoussystem of parameters if and only if

dimk

(k[vi : i ∈ I]/sI(t)

)<∞

for each simplex I ∈ K, where sI(t) is the image of the sequence t under therestriction map sI .

Proof. Assume that t is an hsop. By applying the right exact functor ⊗k[t] tothe epimorphism sI : k[K]→ k[vi : i ∈ I] we obtain that k[K]/t → k[vi : i ∈ I]/sI(t)is also an epimorphism. Hence,

dimk

(k[vi : i ∈ I]/sI(t)

)6 dimk

(k[K]/t

)<∞.

For the opposite statement, assume that

dimk

⊕

I∈K

k[vi : i ∈ I]/sI(t) <∞.

Consider the short exact sequence of k[t ]-modules

0 −→ k[K] ⊕sI−→⊕

I∈K

k[vi : i ∈ I] −→ Q −→ 0,

where Q is the quotient module, and the following fragment of the correspondinglong exact sequence for Tor (Theorem A.2.5 (e)):

· · · −→ Tor−1k[t ](Q,k) −→ k[K]/t −→

⊕

I∈K

k[vi : i ∈ I]/sI(t) −→ · · · .


Since⊕

I∈K k[vi : i ∈ I] is a finitely generated k[t ]-module, its quotient Q is also

finitely generated. Hence dimk Tor−1k[t ](Q,k) <∞ (see Proposition A.2.6), and, by

the exact sequence above, dimk

(k[K]/t

)<∞. Therefore, t is an hsop in k[K].

Recall that we refer to a sequence t = (t1, . . . , tn) ∈ k[K] as linear if deg ti = 2for all i. We may write a linear sequence as

(3.12) ti = λi1v1 + · · ·+ λimvm, for 1 6 i 6 n.

Here is a simple characterisation of lsops in a face ring (see Definition A.3.9):

Lemma 3.3.2. A linear sequence t = (t1, . . . , tn) ⊂ k[K], dimK = n − 1, isan lsop if and only if the restriction sI(t) to each simplex of I ∈ K generates thepolynomial algebra k[vi : i ∈ I] (i.e. the rank of the n × |I| matrix ΛI = (λij),1 6 i 6 n, j ∈ I, is equal to |I|).

Proof. Indeed, if t is linear, then the conditions dimk k[vi : i ∈ I]/sI(t) <∞and k[vi : i ∈ I]/sI(t) ∼= k are equivalent. The latter means that sI(t) generatesk[vi : i ∈ I] as a k-algebra.

Note that it is enough to verify the conditions of Lemmata 3.3.1 and 3.3.2 onlyfor maximal simplices I ∈ K.

Definition 3.3.3. A linear sequence t = (t1, . . . , tn) ⊂ Z[K] is referred toas a integral lsop if its reduction modulo p is an lsop in Zp[K] for any prime p.Equivalently, t is an integral lsop if n = dimK+1 and the restriction sI(t) to eachsimplex I ∈ K generates the polynomial ring Z[vi : i ∈ I] (the equivalence of thesetwo conditions is an exercise).

Although the rational face ring Q[K] always admit an lsop by Theorem A.3.10,an lsop in Zp[K] for a prime p (or an integral lsop in Z[K]) may fail to exist, as isshown by the next example.

Example 3.3.4.1. Let K be a simplicial complex of dimension n− 1 on m > 2n vertices whose

1-skeleton is a complete graph. Then the face ring Z2[K] does not admit an lsop.Indeed, assume that (3.12) is an lsop. Then, by Corollary 3.3.2, each column vectorof the n×m-matrix (λij) is nonzero, and all column vectors are pairwise different(since each pair of vertices of K spans an edge). This is a contradiction, since thenumber of different nonzero vectors in Zn2 is 2n − 1. By considering the reductionmodulo 2 we obtain that Z[K] also does not admit an integral lsop.

2. There are also simple polytopes P whose face rings k[P ] do not admitan lsop over Z2 or Z. Indeed, let P be the dual of a 2-neighbourly simplicial n-polytope (e.g., a cyclic polytope of dimension n > 4, see Example 1.1.17) withm > 2n vertices. Then the 1-skeleton of KP is a complete graph, and thereforeZ[P ] = Z[KP ] does not admit an lsop. This example is taken from [90].

By considering the reduction modulo 2 we observe that the ring Z[K] for K fromthe previous example also does not admit an integral lsop. Existence of integrallsop’s in the face rings Z[K] is a very subtle question of great importance for torictopology; it will be discussed in more detail in Section 4.8.

Definition 3.3.5. K is a Cohen–Macaulay complex over a field k if k[K] is aCohen–Macaulay algebra. We say that K is a Cohen–Macaulay complex over Z, or

3.3. COHEN–MACAULAY COMPLEXES 113

simply a Cohen–Macaulay complex, if k[K] is a Cohen–Macaulay algebra for k = Qand any finite field.

Remark. We shall often consider k[K] as a k[m]-module rather than a k-algebra. However, this does not affect regular sequences and the Cohen–Macaulayproperty: it is an easy exercise to show that a sequence t ⊂ k[m] is k[m]-regularfor k[K] as a k[m]-module if and only the image of t in k[K] is k[K]-regular. Inparticular, k[K] is a Cohen–Macaulay algebra if and only if it is a Cohen–Macaulayk[m]-module. We shall therefore not distinguish between these two notions.

Example 3.3.6. Let K = ∂∆2. Then k[K] = k[v1, v2, v3]/(v1v2v3) andthe Krull dimension is dimk[K] = 2. The elements v1, v2 ∈ k[K] are alge-braically independent, but do not form an lsop, since k[K]/(v1, v2) ∼= k[v3] anddim

(k[K]/(v1, v2)

)= 1. On the other hand, the elements t1 = v1 − v3, t2 = v2 − v3

form an lsop, since k[K]/(t1, t2) ∼= k[t]/t3. It is easy to see that k[K] is a freek[t1, t2]-module on the basis 1, v1, v21. Therefore, k[K] is a Cohen–Macaulay ringand (t1, t2) is a regular sequence.

Cohen–Macaulay complexes can be characterised homologically as follows:

Proposition 3.3.7. The following conditions are equivalent for a simplicialcomplex K of dimension n− 1 with m vertices:

(a) K is Cohen–Macaulay over a field k;(b) β−i(k[K]) = 0 for i > m− n and β−(m−n)(k[K]) 6= 0.

Proof. By definition, condition (a) is that depthk[K] = n. Condition (b)is equivalent to the equality pdimk[K] = m − n, by Corollary A.2.7. Sincedepthk[m] = m, the two conditions are equivalent by Theorem A.3.8.

Example 3.3.8. Let K be the 6-vertex triangulation of RP 2, see Exam-ple 3.2.12.4 and Figure 3.2. Then m− n = 3, and, by Theorem 3.2.4,

β−4(Z2[K]

)= dimZ2 H

1(RP 2;Z2) = 1,

so K is not Cohen–Macaulay over Z2. On the other hand, a similar calculationshows that if the characteristic of k is not 2, then β−i(k[K]) = 0 for i > 3 andβ−3(k[K]) = 6, i.e. K is Cohen–Macaulay over such fields.

Proposition 3.3.9 (Stanley). If K is a Cohen–Macaulay complex of dimensionn− 1, then h(K) = (h0, . . . , hn) is an M -vector (see Definition 1.4.13).

Proof. Let K be Cohen–Macaulay, and let t = (t1, . . . , tn) be an lsop in k[K],where k is a field of zero characteristic. Then, by Proposition A.3.14,

F(k[K], λ

)=F(k[K]/t ;λ

)

(1 − λ2)n .

On the other hand, the Poincare series of k[K] is given by Theorem 3.1.10, whichimplies that

F(k[K]/t ;λ

)= h0 + h1λ

2 + · · ·+ hnλ2n.

Now, A = k[K]/t is a graded algebra generated by its degree-two elements anddimkA

2i = hi, so (h0, . . . , hn) is an M -vector by definition.


Remark. According to a result of Stanley [293, Theorem II.3.3], if (h0, . . . , hn)is an M -vector, then there exists an (n− 1)-dimensional Cohen–Macaulay complexK such that hi(K) = hi. Together with Proposition 3.3.9, this gives a completecharacterisation of face vectors of Cohen–Macaulay complexes.

The following fundamental result gives a combinatorial characterisation ofCohen–Macaulay complexes:

Theorem 3.3.10 (Reisner [276]). Let k = Z or a field. A simplicial complex Kis Cohen–Macaulay over k if and only if for any simplex I ∈ K (including I = ∅)

and i < dim(lk I), we have Hi(lk I;k) = 0.

The proof can be found in many texts on combinatorial commutative algebra,see e.g. [293, §II.4] or [225, Chapter 13]. It is not very hard, but uses the notionof local cohomology, which is beyond the scope of this book.

The following reformulation of Reisner’s Theorem shows that the Cohen–Macaulayness is a topological property of a simplicial complex.

Proposition 3.3.11 (Munkres, Stanley). A simplicial complex K is Cohen–Macaulay over k if and only if for any point x ∈ |K|, we have

Hi(|K|, |K|\ x;k) = Hi(K;k) = 0 for i < dimK.Proof. Let I ∈ K. If I = ∅, then Hi(K;k) = Hi(lk I;k). If I 6= ∅, then

(3.13) Hi(|K|, |K|\ x;k) = Hi−|I|(lk I;k)

for any x in the interior of I, by Proposition 2.2.14.If K is Cohen–Macaulay, then it is pure (Exercise 3.3.20) and therefore lk I is

pure of dimension dimK − |I| (Exercise 2.2.19). Hence, i < dimK implies thati− |I| < dim lk I and Hi(|K|, |K|\ x;k) = 0 by (3.13) and Theorem 3.3.10.

On the other hand, if Hi(|K|, |K|\ x;k) = 0 for i < dimK, then Hj(lk I;k) =Hj+|I|(|K|, |K|\ x;k) = 0 for j < dim lk I, since j + |I| < dim lk I + |I| 6 dimK.Thus, K is Cohen–Macaulay by Theorem 3.3.10.

Corollary 3.3.12. If a triangulation of a space X is a Cohen–Macaulay com-plex, then any other triangulation of X is Cohen–Macaulay as well.

Corollary 3.3.13. Any triangulated sphere is a Cohen–Macaulay complex.

In particular, the h-vector of a triangulated sphere is an M -vector. This factwas used by Stanley in his generalisation of the UBT (Theorem 1.4.4) to arbitrarysphere triangulations:

Theorem 3.3.14 (UBT for spheres). For any triangulated (n− 1)-dimensionalsphere K with m vertices, the h-vector (h0, h1, . . . , hn) satisfies the inequalities

hi(K) 6(m−n+i−1

i

).

Therefore, the UBT holds for triangulated spheres, that is,

fi(K) 6 fi(Cn(m)

)for i = 1, . . . , n− 1.

Proof. Let A = k[K]/t be the algebra from the proof of Proposition 3.3.9, sothat dimkA

2i = hi. In particular, dimkA2 = h1 = m − n. Since A is generated

by A2, the number hi cannot exceed the number of monomials of degree i in m−ngenerators, i.e. hi 6

(m−n+i−1

i

). The rest follows from Lemma 1.4.6.

3.4. GORENSTEIN COMPLEXES 115

Exercises.

3.3.15. If K is a simplicial complex of dimension n− 1, then dimk[K] = n.

3.3.16. Let t be an lsop in k[K]. Then the k-vector space k[K]/t is generatedby monomials vI for I ∈ K. (Hint: prove that vivI = 0 in k[K]/t for any i ∈ [m]and for any maximal simplex I ∈ K, and then use Proposition 3.1.9).

3.3.17. A sequence t ⊂ k[m] is k[m]-regular for k[K] as a k[m]-module if andonly the image of t in k[K] is k[K]-regular.

3.3.18. Let t = (t1, . . . , tn) ⊂ Z[K] be a linear sequence, dimK = n − 1. Thefollowing conditions are equivalent:

(a) the reduction of t modulo p is an lsop in Zp[K] for any prime p;(b) the restriction sI(t) to each simplex I ∈ K generates the polynomial ring

Z[vi : i ∈ I];(c) for each I ∈ K the columns of the n× |I| matrix (λij), 1 6 i 6 n, j ∈ I,

generate the integer lattice Z|I|.

3.3.19. A finitely generated commutative k-algebra is called a complete inter-section algebra if it is the quotient of a polynomial algebra by a regular sequence.Observe that a complete intersection algebra is Cohen–Macaulay. Show that a facering k[K] is a complete intersection algebra if and only if it is isomorphic to thequotient of the form

k[v1, . . . , vm]/(v1v2· · · vk1 , vk1+1vk1+2· · · vk1+k2 , . . . , vk1+···+kp−1+1· · · vk1+···+kp).

This is equivalent to K being decomposable into a join of the form

∂∆k1−1 ∗ ∂∆k2−1 ∗ · · · ∗ ∂∆kp−1 ∗∆m−s−1,

where s = k1 + · · ·+ kp and the join factor ∆m−s−1 is void if s = m.

3.3.20. A Cohen–Macaulay complex is pure. (Hint: given a maximal simplexJ ∈ K, consider the ideal IJ = (vi : i /∈ J) in k[m], and show that

depthk[K] 6 dim(k[m]/IJ) = dimk[vj : j ∈ J ] = |J |.)

3.4. Gorenstein complexes and Dehn–Sommerville relations

Gorenstein rings are a class of Cohen–Macaulay rings with a special dualityproperty. As in the case of Cohen–Macaulayness, simplicial complexes whose facerings are Gorenstein play an important role in combinatorial commutative algebra.In a sense, non-acyclic Gorenstein complexes provide a ‘best possible algebraic ap-proximation’ to sphere triangulations. We review here the most important aspectsof Gorenstein complexes. The proofs of the main results of this section, in particularTheorems 3.4.2 and 3.4.4, require considerably more commutative algebraic tech-niques than those from the previous sections. We refer the reader to [45, Chapter 3]for the general theory of Gorenstein rings and the missing proofs.

We recall from Proposition 3.3.7 that nonzero Betti numbers of a Cohen–Macaulay complex K of dimension n− 1 with m vertices appear up to homologicaldegree −(m− n), and β−(m−n)(k[K]) 6= 0.

Definition 3.4.1. A Cohen–Macaulay complex K of dimension n − 1 withm vertices is called Gorenstein (over a field k) if β−(m−n)(k[K]) = 1, that is, if


Tor−(m−n)k[m] (k[K],k) ∼= k. Furthermore, K is Gorenstein* if K is Gorenstein and

K = coreK (see Definition 2.2.15).

Since K = core(K) ∗∆s−1 for some s, we have k[K] = k[core(K)] ⊗ k[s]. ThenLemma A.3.5 implies that

Tor−ik[m]

(k[K],k

) ∼= Tor−ik[m−s]

(k[coreK],k

).

Therefore, K is Gorenstein if and only if coreK is Gorenstein*.As in the case of Cohen–Macaulay complexes, Gorenstein* complexes can be

characterised topologically as follows.

Theorem 3.4.2 ([293, §II.5] or [45, Theorem 5.6.1]). The following conditionsare equivalent:

(a) K is a Gorenstein* complex over k;(b) for any simplex I ∈ K (including I = ∅) the subcomplex lk I has homology

of a sphere of dimension dim(lk I);(c) for any x ∈ |K|,

Hi(|K|, |K|\ x;k

)= Hi(K;k) =

k if i = dimK;

0 otherwise.

In topology, polyhedra |K| satisfying the conditions of the previous theorem aresometimes called generalised homology spheres (‘generalised’ because a homologysphere is usually assumed to be a manifold). In particular, triangulated spheresare Gorenstein* complexes. Triangulated manifolds are not Gorenstein* or evenCohen–Macaulay in general (Buchsbaum complexes provide a proper algebraic ap-proximation to triangulated manifolds, see [293, § II.8]). Nevertheless, the Tor-algebra of a Gorenstein* complex behaves like the cohomology algebra of a mani-fold: it satisfies Poincare duality. This fundamental result was proved by Avramovand Golod for Noetherian local rings; here we state the graded version of theirtheorem in the case of face rings.

Definition 3.4.3. A graded commutative connected k-algebra A is called a

Poincare algebra if it is finite dimensional over k, i.e. A =⊕d

i=0Ai, and the

k-linear maps

Ai → Homk(Ad−i, Ad),

a 7→ ϕa, where ϕa(b) = ab

are isomorphisms for 0 6 i 6 d. The classical example of a Poincare algebra is thecohomology algebra of a manifold.

Theorem 3.4.4 (Avramov–Golod, [45, Theorem 3.4.5]). A simplicial com-

plex K is Gorenstein* if and only if the algebra T =⊕d

i=0 Ti, where T i =

Tor−ik[m](k[K],k) and d = maxj : Tor−j

k[m](k[K],k) 6= 0, is a Poincare algebra.

Corollary 3.4.5. Let K be a Gorensten* complex of dimension n− 1 on theset [m]. Then the Betti numbers and the Poincare series of the Tor groups satisfy

β−i, 2j(k[K]

)= β−(m−n)+i, 2(m−j)

(k[K]

), 0 6 i 6 m− n, 0 6 j 6 m,

F(Tor−i

k[m](k[K],k); λ)= λ2mF

(Tor

−(m−n)+ik[m] (k[K],k); 1

λ

).

3.4. GORENSTEIN COMPLEXES 117

Proof. Theorems 3.2.4 and 3.4.2 imply that

β−(m−n)(k[K]

)= β−(m−n),2m

(k[K]

)= 1.

We therefore have d = m − n and T d = Tor−(m−n), 2mk[m] (k[K],k) ∼= k in the nota-

tion of Theorem 3.4.4. Since the multiplication in the Tor-algebra preserves the

bigrading, the isomorphisms T i∼=→ Homk(T

m−n−i, Tm−n) from the definition of aPoincare algebra can be refined to isomorphisms

T i, 2j∼=−→ Homk(T

m−n−i, 2(m−j), Tm−n, 2m),

where Tm−n, 2m ∼= k. This implies the first identity, and the second is a directcorollary.

As a further corollary we obtain the following symmetry property for thePoincare series of the face ring:

Corollary 3.4.6. If K is Gorenstein* of dimension n− 1, then

F(k[K], λ

)= (−1)nF

(k[K], 1

λ

).

Proof. We apply Proposition A.2.1 to the minimal resolution of the k[m]-module k[K]. Note that F (k[m];λ) = (1 − λ2)−m. It follows from the formula forthe Poincare series from Proposition A.2.1 and Proposition A.2.6 that

F(k[K];λ

)= (1− λ2)−m

m−n∑

i=0

(−1)iF(Tor−i

k[m]

(k[K],k

);λ).

Using Corollary 3.4.5, we calculate

F(k[K];λ

)= (1− λ2)−m

m−n∑

i=0

(−1)iλ2mF(Tor

−(m−n)+ik[m]

(k[K],k

); 1λ

)=

=(1− ( 1λ)

2)−m

(−1)mm−n∑

j=0

(−1)m−n−jF(Tor−j

k[m]

(k[K],k

); 1λ

)=

= (−1)nF(k[K]; 1

λ

).

Corollary 3.4.7. The Dehn–Sommerville relations hi = hn−i hold for anyGorenstein* complex of dimension n− 1 (in particular, for any triangulation of an(n− 1)-sphere).

Proof. This follows from the explicit form of the Poincare series for k[K](Theorem 3.1.10) and the previous corollary.

The Dehn–Sommerville relations may be further generalised to wider classes ofcomplexes and posets; we give some of these generalisations in Section 3.8 below.

Unlike the situation with Cohen–Macaulay complexes, a characterisation ofh-vectors (or, equivalently, f -vectors) of Gorenstein complexes is not known:

Problem 3.4.8 (Stanley). Characterise the h-vectors of Gorenstein complexes.

If the g-conjecture (i.e. the inequalities of Theorem 1.4.14 (b) and (c)) holdsfor Gorenstein complexes, then this would imply a solution to the above problem.


Exercises.

3.4.9. A Gorenstein complex K is Gorenstein* if and only if it is non-acyclic

(i.e. H∗(K;k) 6= 0).

3.4.10 ([293, Theorem II.5.1]). Show that K is Gorenstein* if and only if thefollowing three conditions are satisfied:

(a) K is Cohen–Macaulay;(b) every (n− 2)-dimensional simplex is contained in exactly two (n− 1)-di-

mensional simplices;(c) χ(K) = χ(Sn−1), where χ( · ) denotes the Euler characteristic.

3.4.11. Let K be a Gorenstein* complex of dimension n− 1 on [m]. Show that

Hk(KJ ) ∼= Hn−2−k(KJ )for any J ⊂ [m], where J = [m] \ J . This is known as Alexander duality fornon-acyclic Gorenstein complexes .

3.5. Face rings of simplicial posets

The whole theory of face rings may be extended to simplicial posets (definedin Section 2.8), thereby leading to new important classes of rings in combinatorialcommutative algebra and applications in toric topology.

The face ring k[S] of a simplicial poset S was introduced by Stanley [292] as aquotient of a certain graded polynomial ring by a homogeneous ideal determined bythe poset relation in S. The rings k[S] have remarkable algebraic and homologicalproperties, albeit they are much more complicated than the Stanley–Reisner facerings k[K]. Unlike k[K], the ring k[S] is not generated in the lowest positive degree.Face rings of simplicial posets were further studied by Duval [101] and Maeda–Masuda–Panov [209], [204], among others. Cohen–Macaulay and Gorenstein* facerings are particularly important; both properties are topological, that is, dependonly on the topological type of the geometric realisation |S|.

As usual, we shall not distinguish between simplicial posets S and their geo-metric realisations (simplicial cell complexes) |S|. Given two elements σ, τ ∈ S, wedenote by σ ∨ τ the set of their joins, and denote by σ ∧ τ the set of their meets.Whenever either of these sets consists of a single element, we use the same notationfor this particular element of S.

To make clear the idea behind the definition of the face ring of a simplicialposet, we first consider the case when S is a simplicial complex K. Then σ ∧ τconsists of a single element (possibly ∅), and σ ∨ τ is either empty or consistsof a single element. We consider the graded polynomial ring k[vσ : σ ∈ K] withone generator vσ of degree deg vσ = 2|σ| for each simplex σ ∈ K. The followingproposition provides an alternative presentation of the face ring k[K], with a largerset of generators:

Proposition 3.5.1. There is a canonical isomorphism of graded rings

k[K] ∼= k[vσ : σ ∈ K]/I ′K,where I ′K is the ideal generated by the element v∅ − 1 and all elements of the form

vσvτ − vσ∧τvσ∨τ .Here we set vσ∨τ = 0 whenever σ ∨ τ is empty.

3.5. FACE RINGS OF SIMPLICIAL POSETS 119

σ

21

τ

σ

(a) r = 2 (b) r = 3

1

3

2

τ

e

Figure 3.3. Simplicial cell complexes.

Proof. The isomorphism is established by the map taking vσ to∏i∈σ vi. The

rest is left as an exercise.

Now let S be an arbitrary simplicial poset with the vertex set V (S) = [m]. Inthis case both σ ∨ τ and σ ∧ τ may consist of more than one element, but σ ∧ τconsists of a single element whenever σ ∨ τ is nonempty.

We consider the graded polynomial ring k[vσ : σ ∈ S] with one generator vσ ofdegree deg vσ = 2|σ| for every element σ ∈ S.

Definition 3.5.2 ([292]). The face ring of a simplicial poset S is the quotient

k[S] = k[vσ : σ ∈ S] / IS ,where IS is the ideal generated by the elements v0 − 1 and

(3.14) vσvτ − vσ∧τ ·∑

η∈σ∨τ

vη.

The sum over the empty set is zero, so we have vσvτ = 0 in k[S] if σ ∨ τ is empty.The grading may be refined to an Nm-grading by setting mdeg vσ = 2V (σ).

Here V (σ) ⊂ [m] is the vertex set of σ, and we identify subsets of [m] with (0, 1)-vectors in 0, 1m ⊂ Nm as usual. In particular, mdeg vi = 2ei.

Example 3.5.3.1. The simplicial cell complex shown in Fig. 3.3 (a) is obtained by gluing two

segments along their boundaries and has rank 2. The vertices are 1, 2 and we denotethe 1-dimensional simplices by σ and τ . Then the face ring k[S] is the quotient ofthe graded polynomial ring

k[v1, v2, vσ, vτ ], deg v1 = deg v2 = 2, deg vσ = deg vτ = 4

by the two relationsv1v2 = vσ + vτ , vσvτ = 0.

2. The simplicial cell complex in Fig. 3.3 (b) is obtained by gluing two trianglesalong their boundaries and has rank 3. The vertices are 1, 2, 3 and we denote the1-dimensional simplices (edges) by e, f and g, and the 2-dimensional simplices byσ and τ . The face ring k[S] is isomorphic to the quotient of the polynomial ring

k[v1, v2, v3, vσ, vτ ], deg v1 = deg v2 = deg v3 = 2, deg vσ = deg vτ = 6

by the two relationsv1v2v3 = vσ + vτ , vσvτ = 0.

The generators corresponding to the edges can be excluded because of the relationsve = v1v2, vf = v2v3 and vg = v1v3.


Remark. The ideal IS is generated by straightening relations (3.14); theserelations allow us to express the product of any pair of generators via productsof generators corresponding to pairs of ordered elements of the poset. This canbe restated by saying that k[S] is an example of an algebra with straightening law(ASL for short, also known as a Hodge algebra). Lemma 3.5.4 and Theorem 3.5.7below reflect algebraic properties of ASL’s, and may be restated in this generality.For more on the theory of ASL’s see [293, § III.6] and [45, Chapter 7].

A monomial vi1σ1vi2σ2· · · vikσk

∈ k[vσ : σ ∈ S] is standard if σ1 < σ2 < · · · < σk.

Lemma 3.5.4. Any element of k[S] can be written as a linear combination ofstandard monomials.

Proof. It is enough to prove the statement for elements of k[S] representedby monomials in generators vσ. We write such a monomial as a = vτ1vτ2 · · · vτkwhere some of the τi may coincide. We need to show that any such monomial canbe expressed as a sum of monomials

∑vσ1 · · · vσl

with σ1 6 · · · 6 σl. We mayassume by induction that τ2 6 · · · 6 τk. Using relation (3.14) we can replace a by

vτ1∧τ2

( ∑

ρ∈τ1∨τ2

vρ

)vτ3 · · · vτk .

Now the first two factors in each summand above correspond to ordered elementsof S. We proceed by replacing the products vρvτ3 by vρ∧τ3(

∑π∈ρ∨τ3

vπ). Sinceτ1∧τ2 6 ρ∧τ3, now the first three factors in each monomial are in order. Continuingthis process, we obtain in the end a sum of monomials corresponding to totallyordered sets of elements of S.

We refer to the presentation from Lemma 3.5.4 as a standard representation ofan element a ∈ k[S].

Given σ ∈ S, we define the corresponding restriction homomorphism as

sσ : k[S]→ k[S]/(vτ : τ 66 σ).

The following result is straightforward.

Proposition 3.5.5. Let |σ| = k with V (σ) = i1, . . . , ik. Then the image ofthe homomorphism sσ is the polynomial ring k[vi1 , . . . , vik ].

The next result generalises Proposition 3.1.8 to simplicial posets.

Theorem 3.5.6. The direct sum

s =⊕

σ∈Ssσ : k[S] −→⊕

σ∈S

k[vi : i ∈ V (σ)]

of all restriction maps is a monomorphism.

Proof. Take a nonzero element a ∈ k[S] and write its standard representation.Fix a standard monomial vi1σ1

· · · vikσkwhich enters into this decomposition with a

nonzero coefficient. Allowing some of the exponents ij to be zero, we may assumethat σk is a maximal element in S and |σj | = j for 1 6 j 6 k. We shall provethat sσk

(a) 6= 0. Identify sσk(k[S]) with the polynomial ring k[t1, . . . , tk] (so that

tj = vij in the notation of Proposition 3.5.5). Then sσk(vσk

) = t1 · · · tk, and we mayassume without loss of generality that sσk

(vσj) = t1 · · · tj for 1 6 j 6 k. Hence,

sσk

(vi1σ1· · · vikσk

)= ti11 (t1t2)

i2 · · · (t1 · · · tk)ik .


If we prove that no other monomial vj1τ1 · · · vjmτm is mapped by sσkto the same element

of k[t1, . . . , tk], then this would imply that sσk(a) 6= 0. Note that

sσk(vj1τ1 · · · vjmτm) = 0 if τi 66 σk for some i with ji 6= 0,

so that we may assume that m = k. Now suppose that

(3.15) sσk

(vi1σ1· · · vikσk

)= sσk

(vj1τ1 · · · vjkτk

).

We shall prove that vi1σ1· · · vikσk

= vj1τ1 . . . vjkτk . We may assume by induction that the

‘tails’ of these monomial coincide, that is, there is some q, 1 6 q 6 k, such thatip = jp and σp = τp for ip 6= 0 whenever p > q. We shall prove that iq = jq andσq = τq if iq 6= 0. We obtain from (3.15) that

sσk

(vi1σ1· · · viqσq

)(t1 · · · tq+1)

iq+1 · · · (t1 · · · tk)ik =

= sσk

(vj1τ1 · · · vjqτq

)(t1 · · · tq+1)

iq+1 · · · (t1 · · · tk)jk ,

hence, sσk(vi1σ1· · · viqσq ) = sσk

(vj1τ1 · · · vjqτq ). Let jl be the last nonzero exponent in

vj1τ1 · · · vjqτq (i.e. jl+1 = · · · = jq = 0). Then we also have il+1 = · · · = iq = 0,

as otherwise sσk(vi1σ1· · · viqσq ) is divisible by t1 · · · tl+1, while sσk

(vj1τ1 · · · vjqτq ) is not.

We also have il = jl and σl = τl, since il is the maximal power of the monomial

t1 · · · tl which divides sσk(vi1σ1· · · viqσq ). We conclude by induction that vi1σ1

· · · vikσk=

vj1τ1 · · · vjkτk , and sσk(a) 6= 0.

Remark. The proof above also shows that the map s =⊕

σ sσ in Theo-rem 3.5.6 can be defined as the sum over only the maximal elements σ ∈ S.

Theorem 3.5.7. The standard representation of an element a ∈ k[S] is unique.In other words, the standard monomials vi1σ1

· · · vikσkform a k-basis of k[S].

Proof. This follows directly from Lemma 3.5.4 and Theorem 3.5.6.

Lemma 3.3.1, which describes hsop’s in the face rings of simplicial complexes,can be readily extended to simplicial posets (the same proof based on the propertiesof the restriction map s works):

Lemma 3.5.8. Let S be a simplicial poset of rank n. A sequence of homogeneouselements t = (t1, . . . , tn) of k[S] is a homogeneous system of parameters if and only

dimk

(k[vi : i ∈ V (σ)]/sσ(t)

)<∞

for each element σ ∈ S.The f -vector of a simplicial poset S is f (S) = (f0, . . . , fn−1), where n − 1 =

dimS and fi is the number of elements of rank i + 1 (i.e. the number of faces ofdimension i in the simplicial cell complex). We also set f−1 = 1. The h-vectorh(S) = (h0, . . . , hn) is then defined by (2.3).

The Poincare series of the face ring k[S] has exactly the same form as in thecase of simplicial complexes:

Theorem 3.5.9. We have

F(k[S];λ

)=

n∑

k=0

fk−1λ2k

(1− λ2)k =h0 + h1λ

2 + · · ·+ hnλ2n

(1− λ2)n .


Proof. By Theorem 3.5.7, we need to calculate the Poincare series of the k-vector space generated by the monomials vi1σ1

· · · vikσkwith σ1 < · · · < σk. For every

σ ∈ S denote byMσ the set of such monomials with σk = σ and ik > 0. Let |σ| = k;consider the restriction homomorphism sσ to the polynomial ring k[t1, . . . , tk]. Thensσ(Mσ) is the set of monomials in k[t1, . . . , tk] which are divisible by t1 · · · tk.Therefore, the Poincare series of the subspace generated by the setMσ is λ2k

(1−λ2)k .

Now, to finish the proof of the first identity we note that S is the union⋃σ∈SMσ

of the nonintersecting subsets Mσ. The second identity follows from (2.3).

As we have seen in Exercise 3.1.15, the face ring k[K] of a simplicial complexcan be realised as the limit of a diagram of polynomial algebras over catop(K). Asimilar description exists for the face ring k[S]:

Construction 3.5.10 (k[S] as a limit). We consider the diagram k[ ·]S similarto that of Exercise 3.1.15:

k[ · ]S : catop(S) −→ cga,

σ 7−→ k[vi : i ∈ V (σ)],

whose value on a morphism σ 6 τ is the surjection

k[vi : i ∈ V (τ)]→ k[vi : i ∈ V (σ)]

sending each vi with i /∈ V (σ) to zero.

Lemma 3.5.11. We have

k[S] = limk[ · ]S

where the limit is taken in the category cga.

Proof. We set up a total order on the elements of S so that the rank func-tion does not decrease, and proceed by induction. We therefore may assume thestatement is proved for a simplicial poset T , and need to prove it for S which isobtained from T by adding one element σ. Then S<σ = τ ∈ S : τ < σ is the faceposet of the boundary of the simplex ∆σ. Geometrically, we may think of |S| asobtained from |T | by attaching one simplex ∆σ along its boundary (if |σ| = 1, then∆σ is a single point, so |S| is a disjoint union of |T | and a point). We thereforeneed to prove that the following is a pullback diagram:

(3.16)

k[S] −−−−→ k[S6σ]yy

k[T ] −−−−→ k[S<σ ].Here the vertical arrows map vσ to 0, while the horizontal ones map vτ to 0 forτ 66 σ. Denote by A the pullback of (3.16) with k[S] dropped. We need to showthat the natural map k[S]→ A is an isomorphism.

Since the limits in cga are created in the underlying category of graded k-vector spaces, the space of A is the direct sum of k[T ] and k[S6σ ] with the piecesk[S<σ] identified in both spaces. In other words,

(3.17) A = T ⊕ k[S<σ]⊕ S,where T is the complement to k[S<σ] in k[T ], and S is the complement to k[S<σ]in k[S6σ ]. By Theorem 3.5.7, the space k[S<σ] has basis of standard monomials


vj1τ1vj2τ2 · · · vjkτk with τk < σ. Similarly, S has basis of those monomials with τk = σ and

jk > 0, while T has basis of those monomials with τk 66 σ and jk > 0. Yet anotherapplication of Theorem 3.5.7 gives a decomposition of k[S] identical to (3.17): astandard basis monomial vj1τ1v

j2τ2 · · · vjkτk with jk > 0 has either τk 66 σ, or τk < σ, or

τk = σ. These three possibilities map to T , k[S<σ] and S respectively. It followsthat k[S]→ A is an isomorphism of k-vector spaces. Since it is an algebra map, itis also an isomorphism of algebras, thus finishing the proof.

The description of k[S] as a limit has the following important corollary, describ-ing the functorial properties of the face rings and generalising Proposition 3.1.5.

Proposition 3.5.12. Let f : S → T be a rank-preserving map of simplicialposets. Define a homomorphism

f∗ : k[wτ : τ ∈ T ]→ k[vσ : σ ∈ S], f∗(wτ ) =∑

σ∈f−1(τ), |σ|=|τ |

vσ.

Then f∗ descends to a ring homomorphism k[T ] → k[S], which we continue todenote by f∗.

Proof. The poset map f gives rise to a functor f : catop(S)→ catop(T ) andtherefore to a natural transformation

f∗ : [catop(T ),cga]→ [catop(S),cga],

where [catop(S),cga] denotes the functors from catop(S) to cga. It is easy tosee that f∗

k[ · ]T = k[ · ]S in the notation of Construction 3.5.10, so we have theinduced map of limits f∗ : k[T ]→ k[S]. We also have that f∗(wτ ) =

∑σ∈f−1(τ) vσ

by the construction of lim in cga.

Example 3.5.13. The folding map (2.9) induces a monomorphism k[KS ] →k[S], which embeds k[KS ] in k[S] as the subring generated by the elements vi.

Remark. An attempt to prove Proposition 3.5.12 directly from the definition,by showing that f∗(IT ) ⊂ IS , runs into a complicated combinatorial analysis ofthe poset structure. This is an example of a situation where the use of an abstractcategorical description of k[S] proves to be beneficial.

Let k[m] = k[v1, . . . , vm] be the polynomial algebra on m generators of degree 2corresponding to the vertices of S. The face ring k[S] acquires a k[m]-algebrastructure via the map k[m] → k[S] sending each vi identically. (Unlike the caseof simplicial complexes, this map is generally not surjective.) We thereby obtain aZ⊕ Nm-graded Tor-algebra of k[S]:

Tork[v1,...,vm]

(k[S],k

)=

⊕

i>0,a∈Nm

Tor−i, 2ak[v1,...,vm]

(k[S],k

),

by analogy with Construction 3.2.8 for simplicial complexes.We finish this section by stating a generalisation of Hochster’s theorem to

simplicial posets, and deriving some of its corollaries.

Theorem 3.5.14 (Duval [101], see also [199]). For any subset J ⊂ [m] wehave


(k[S],k

) ∼= H |J|−i−1(|SJ |;k),where SJ the subposet of S consisting of those σ for which V (σ) ⊂ J . Also,

Tor−i,2ak[m] (k[S],k) = 0 if a is not a (0, 1)-vector.


Proof. The argument follows the lines of the proof of Theorem 3.2.9. Wedefine the quotient differential graded algebra

R∗(S) = Λ[u1, . . . , um]⊗ k[S]/IRwhere IR is the ideal generated by the elements

uivσ with i ∈ V (σ), and vσvτ with σ ∧ τ 6= 0.

Note that the latter condition is equivalent to V (σ) ∩ V (τ) 6= ∅.Then we need to prove the analogue of Lemma 3.2.6, that is, to show that the

quotient projection

: Λ[u1, . . . , um]⊗ k[S] → R∗(S)induces an isomorphism in cohomology. This can be done by providing the appro-priate chain homotopy, as in the proof of Lemma 3.2.6, but the formulae will bemore complicated. Alternatively, we can use a topological argument, see the proofof Theorem 4.10.6 below.

Let Cp−1(|SJ |) denote the (p − 1)th cellular cochain group of |SJ | with coeffi-cients in k. It has a basis of cochains ασ corresponding to elements σ ∈ SJ with|σ| = p. We define a k-linear map

f : Cp−1(|SJ |) −→ Rp−|J|,2J(S), ασ 7−→ ε(V (σ), J)uJ\V (σ)vσ,

where ε(V (σ), J) is the sign from the proof of Theorem 3.2.9. This map is anisomorphism of cochain complexes; the details are left to the reader. Therefore,

Hp−1(|SJ |) ∼= Torp−|J|,2Jk[v1,...,vm]

(k[S],k

),

which is equivalent to the first required isomorphism. Since R−i,2a (S) = 0 if a is

not a (0, 1)-vector, Tor−i,2ak[m] (k[K],k) vanishes for such a .

We define the multigraded algebraic Betti numbers of k[S] as

β−i, 2a(k[S]

)= dimk Tor

−i, 2ak[v1,...,vm]

(k[S],k

),

for 0 6 i 6 m, a ∈ Nm. We also set

β−i(k[S]

)= dimkTor

−ik[v1,...,vm]

(k[S],k

)=∑

a∈Nm

β−i, 2a(k[S]

).

Example 3.5.15. Let S be the simplicial poset of Example 3.5.3.1. By The-orem 3.5.14, β0,(0,0)(k[S]) = β0,(2,2)(k[S]) = 1, and the other Betti numbers arezero. This implies that k[S] is a free k[v1, v2]-module with two generators, 1 and vσ,of degree 0 and 4 respectively.

Note that unlike the case of simplicial complexes, β0(k[S]) may be larger than 1.In fact, the following proposition follows easily from Theorem 3.5.14.

Proposition 3.5.16. The number of generators of k[S] as a k[m]-module equals

β0(k[S]) =∑

J⊂[m]

dim H |J|−1(|SJ |

).


Exercises.

3.5.17. Finish the proof of Proposition 3.5.1.

3.5.18. Fill in the details in the proof of Theorem 3.5.14.

3.5.19. Calculate the multigraded Betti numbers for the simplicial poset ofExample 3.5.3.2.

Face rings: additional topics

3.6. Cohen–Macaulay simplicial posets

Assume given a property A of simplicial complexes. Then we can extend thisproperty to posets by postulating that a poset P has the property A if the ordercomplex ord(P) (see Definition 2.3.6) has the property A. In particular, Cohen–Macaulay and Gorenstein posets can be defined in this way. Simplicial posets Sare of particular interest to us; in this case the order complex is identified withthe barycentric subdivision S ′ (to be precise, with the cone over the barycentricsubdivision, as we include the empty simplex, but this difference is inessential forthe definitions to follow).

Definition 3.6.1. A simplicial poset S is Cohen–Macaulay (over k) if itsbarycentric subdivision S ′ is a Cohen–Macaulay simplicial complex.

By definition, S is a Cohen–Macaulay simplicial poset if and only if the face ringk[S ′] is Cohen–Macaulay. Since the face ring is also defined for the face poset S itself(and not only for its barycentric subdivision), it is perfectly natural to ask whetherthe class of Cohen–Macaulay simplicial posets admits an intrinsic description interms of their face rings k[S]. One would achieve such a description by proving thatthe ring k[S ′] is Cohen–Macaulay if and only if the ring k[S] is Cohen–Macaulay.The ‘if’ part follows from the general theory of ASL’s, see [292, Corollary 3.7].The ‘only if’ part was proved in [204]; the proof uses the decomposition of thebarycentric subdivision into a sequence of stellar subdivisions and then goes on toshow that the Cohen–Macaulay property is preserved under stellar subdivisions.We include this characterisation of Cohen–Macaulay simplicial posets in terms oftheir face rings in Theorem 3.6.7 below.

Since many of the constructions in this section are geometric, we often talkabout simplicial cell complexes rather than simplicial posets. We say that a simpli-cial subdivision of a simplicial cell complex S is regular if it is a simplicial complex.For instance, the barycentric subdivision is regular. Since the Cohen–Macaulaynessof a simplicial complex is a topological property (see Proposition 3.3.11), we havethe following statement.

Proposition 3.6.2. The following conditions are equivalent:

(a) the barycentric subdivision of a simplicial cell complex S is a Cohen–Macaulay complex;

(b) any regular subdivision of S is a Cohen–Macaulay complex;(c) a regular subdivision of S is a Cohen–Macaulay complex.

As a further corollary we obtain that Proposition 3.3.11 itself extends to sim-plicial cell complexes, i.e. the property of a simplicial cell complex to be Cohen–Macaulay is also topological.

127

128 FACE RINGS: ADDITIONAL TOPICS

By analogy with Definition 2.2.13, we define the star and the link of σ ∈ S asthe following subcomplexes:

stS σ = τ ∈ S : σ ∨ τ is nonempty;lkS σ = τ ∈ S : σ ∨ τ is nonempty, and τ ∧ σ = 0.

Remark. If S is a simplicial complex, then the poset lkS σ is isomorphic tothe open semi-interval

S>σ = ρ ∈ S : ρ > σ,and | stS σ| ∼= ∆σ ∗ | lkS σ|, where ∗ denotes the join. However, none of theseisomorphisms holds for general S, see Example 3.6.5 below.

Because of this remark, we cannot simply extend the definition of stellar subdi-visions (Definition 2.7.1) to simplicial cell complexes. Instead, we define the stellarsubdivision ssσ S of S at σ as the simplicial cell complex obtained by stellarlysubdividing each face containing σ in a compatible way.

Proposition 3.6.3. The barycentric subdivision S ′ can be obtained as a se-quence of stellar subdivisions, one at each face σ ∈ S, starting from the maximalfaces. Moreover, each stellar subdivision in the sequence is applied to a face whosestar is a simplicial complex.

Proof. Assume dimS = n− 1. We start by applying to S stellar subdivisionsat all (n− 1)-dimensional faces. Denote the resulting complex by S1. The (n− 2)-faces of S1 are of two types: the “old” ones, remaining from S, and the “new” ones,appearing as the result of the stellar subdivisions. Then we take stellar subdivisionsof S1 at all “old” (n − 2)-faces, and denote the result by S2. Next we apply to S2stellar subdivisions at all (n−3)-faces remaining from S. Proceeding in this way, atthe end we get Sn−1 = S ′. To prove the second statement, consider two subsequent

complexes R and R in the sequence, so that R is obtained from R by a singlestellar subdivision at some σ ∈ S. Then stR σ is isomorphic to ∆σ ∗ (S>σ)′ andtherefore it is a simplicial complex.

We proceed with two lemmata necessary to prove our main result.

Lemma 3.6.4. Let S be a simplicial poset of rank n with vertex set V (S) = [m],and assume that the first k vertices span a face σ. Assume further that stS σ is a

simplicial complex, and let S be the stellar subdivision of S at σ. Let v denote the

degree-two generator of k[S] corresponding to the added vertex. Then there exists

a unique homomorphism β : k[S]→ k[S] such that

vτ 7→ vτ for τ /∈ stS σ;

vi 7→ v + vi, for i = 1, . . . , k;

vi 7→ vi, for i = k + 1 . . . ,m.

Moreover, β is injective, and if t is an hsop in k[S], then β(t) is an hsop in k[S].Proof. In order to define the map β we first need to specify the images of vτ

for all τ ∈ stS σ. Choose such a vτ and let V (τ) = i1, . . . , iℓ be its vertex set.Then we have the following identity in the ring k[S] = k[vτ : τ ∈ S]/IS :

(3.18) vi1 · · · viℓ = vτ +∑

η : V (η)=V (τ), η 6=τ

vη.

3.6. COHEN–MACAULAY SIMPLICIAL POSETS 129

For any vη in the latter sum we have η /∈ stS σ, since stS σ is a simplicial complex,in which any set of vertices spans at most one face. Since β is already defined onthe product on the left hand side and on the sum on the right hand side above, thisdetermines β(vτ ) uniquely.

We therefore obtain a map of polynomial algebras k[vτ : τ ∈ S]→ k[vτ : τ ∈ S](which we denote by the same letter β for a moment), and need to check that it

descends to a map of face rings, k[S] → k[S]. In other words, we need to verifythat β(IS) ⊂ IS .

It is clear from the definition of β that we have the commutative diagram

k[vτ : τ ∈ S]p //

β

k[S] s //

β

⊕τ∈S k[vi : i ∈ V (τ)]

s(β)

k[vτ : τ ∈ S]

p // k[S] s //⊕τ∈S k[vi : i ∈ V (τ)],

in which the middle vertical map is not yet defined. Here by s and s we denotethe restriction maps from Theorem 3.5.6, and s(β) is the map induced by β on thedirect sum of polynomial algebras. Now let x ∈ IS , i.e. p(x) = 0. Then, by thecommutativity of the diagram, spβ(x) = 0. Since s is injective, we have pβ(x) = 0.Hence, β(x) ∈ IS , which implies that the middle vertical map is well defined.

The last statement also follows from the commutative diagram above. Themap s(β) sends each direct summand of its domain isomorphically to at least onesummand of its range, and therefore it is injective. Thus, β : k[S] → k[S ′] is alsoinjective. The statement about hsop’s then follows Lemma 3.5.8.

Remark. If we defined the map β by sending each vi identically, then it would

still give rise to a ring homomorphism k[S] → k[S], but the latter would not beinjective (for example, it would map vσ ∈ k[S] to zero).

Example 3.6.5. The assumption on stS σ in Lemma 3.6.4 is not always sat-isfied. For example, if S is obtained by identifying two 2-simplices along theirboundaries, and σ is any edge, then stS σ = S, which is not a simplicial complex.

Note also that if stS σ is not a simplicial complex, then the map β : k[S]→ k[S ′]is not determined uniquely by the conditions specified in Lemma 3.6.4 (we cannotdetermine the images of vτ with τ ∈ stS σ). Nevertheless, it is still possible todefine the map β : k[S]→ k[S ′] for an arbitrary simplicial poset S, see Section 7.9.

Lemma 3.6.6. Assume that k[S] is a Cohen–Macaulay ring, and let S be a

stellar subdivision of S at σ such that stS σ is a simplicial complex. Then k[S] is aCohen–Macaulay ring.

Proof. We first prove that stS σ is a Cohen–Macaulay complex. Since stS σ =∆σ ∗ lkS σ, it is enough to verify that lkS σ is Cohen–Macaulay. This follows fromReisner’s Theorem (Theorem 3.3.10) and the fact that simplicial cohomology oflkS σ is a direct summand in local cohomology of k[S] (see [293, Theorem II.4.1]or [45, Theorem 5.3.8]).

Choose an hsop t = (t1, . . . , tn) in k[S] and set t = β(t). Let

p : k[S]→ k[S]/(vτ : τ /∈ stS σ) = k[stS σ]


be the quotient projection. Set R = ker p. Similarly, set

R = ker(p : k[S]→ k[stS v]

),

where v is the new vertex added in the process of stellar subdivision. Since the

simplicial cell complexes S and S do not differ on the complements of stS σ and

stS v respectively, the map β restricts to the identity isomorphism R → R. Wetherefore have the following commutative diagram with exact rows:

0 −−−−→ R −−−−→ k[S] p−−−−→ k[stS σ] −−−−→ 0y∼=

yβy

0 −−−−→ R −−−−→ k[S] p−−−−→ k[stS v] −−−−→ 0,

Applying the functors ⊗k[t ]k and ⊗k[t ]k to the diagram above, we get a map

between the long exact sequences for Tor. Consider the following fragment:

Tor−2k[t ](k[st σ],k)

f→ Tor−1k[t ](R,k)→ Tor−1

k[t ] k[S],k)→ Tor−1k[t ](k[st σ],k)

↓ ↓ ∼= ↓ ↓Tor−2

k[t ](k[st v],k)

f→ Tor−1

k[t ](R,k)→ Tor−1

k[t ](k[S],k)→ Tor−1

k[t ](k[st v],k).

Since k[S] is Cohen–Macaulay, Tor−1k[t ](k[S],k) = 0 and the map f is surjective.

Then f is also surjective. Since stS σ is a Cohen–Macaulay simplicial complex and| stS σ| ∼= | stS v|, Proposition 3.3.11 implies that k[st v] is Cohen–Macaulay. There-

fore, Tor−1

k[t ](k[st v],k) = 0. Since f is surjective, we also have Tor−1

k[t ](k[S],k) = 0.

Hence k[S] is free as a k[t ]-module (see [203, Lemma VII.6.2]) and thereby isCohen–Macaulay.

Now we can prove the main result of this section:

Theorem 3.6.7. A simplicial poset S is Cohen–Macaulay if and only if theface ring k[S] is Cohen–Macaulay.

Proof. The fact that the face ring of a Cohen–Macaulay simplicial poset S isCohen–Macaulay is proved in [292, Corollary 3.7] (see also [293, § III.6]) using thetheory of ASL’s.

Assume now that k[S] is a Cohen–Macaulay ring. Since the barycentric subdi-vision S ′ is obtained by a sequence of stellar subdivisions, subsequent applicationof Lemma 3.6.6 shows that k[S ′] is also Cohen–Macaulay. Thus, S ′ is a Cohen–Macaulay poset.

We end this section by giving Stanley’s characterisation of h-vectors of Cohen–Macaulay simplicial posets.

Theorem 3.6.8 (Stanley). The integer vector h = (h0, h1, . . . , hn) is the h-vector of a Cohen–Macaulay simplicial poset if and only if h0 = 1 and hi > 0.

Proof. Let h = h(S) for a Cohen–Macaulay simplicial poset S. The conditionh0 = 1 follows from the definition of the h-vector, see (2.3). Let k be a field of zerocharacteristic, and t = (t1, . . . , tn) an lsop in k[S] (since k[S] is not generated bylinear elements, the existence of an lsop is not automatic and is left as an exercise;

3.7. GORENSTEIN SIMPLICIAL POSETS 131

alternatively, see [292, Lemma 3.9]). Comparing the formula for the Poincare seriesfrom Proposition A.3.14 with that of Theorem 3.5.9, we obtain

F(k[S]/t ;λ

)= h0 + h1λ

2 + · · ·+ hnλ2n.

Hence, hi > 0, as needed.Now we construct a Cohen–Macaulay simplicial cell complex S with any

given h-vector such that h0 = 1 and hi > 0. First note that h(∆n−1) =(1, 0, . . . , 0) and ∆n−1 is a Cohen–Macaulay simplicial (cell) complex. Now, givenan (n−1)-dimensional Cohen–Macaulay simplicial cell complex S with the h-vector(h0, . . . , hn), it suffices to construct, for any k = 1, . . . , n, a new Cohen–Macaulaysimplicial cell complex Sk with the h-vector given by

(3.19) h(Sk) = (h0, . . . , hk−1, hk + 1, hk+1, . . . , hn).

To do this, we choose an (n− 1)-face of S, and in this face choose some k faces ofdimension n− 2. Then add to S a new (n− 1)-simplex by attaching it along somek faces of dimension n − 2 to the chosen k faces of S. A direct check shows thatthe h-vector of the resulting simplicial cell complex Sk is given by (3.19). The factthat Sk is Cohen–Macaulay follows directly from Proposition 3.3.11.

Note that this characterisation is substantially simpler than that for simplicialcomplexes (see Propositions 3.3.9 and the remark after it).

Exercises.

3.6.9. The map of face rings k[S]→ k[S] of Lemma 3.6.4 is not induced by any

poset map S → S.

3.6.10. Let S be a stellar subdivision of S at σ such that stS σ is a simplicial

complex. Show that the ring k[S] is Cohen–Macaulay if and only if k[S] is Cohen–Macaulay, i.e. the converse of Lemma 3.6.6 holds.

3.6.11. If k is of characteristic zero, then k[S] admits an lsop.

3.7. Gorenstein simplicial posets

Gorenstein simplicial posets arise in toric topology as the combinatorial struc-tures associated to the orbit quotients of torus manifolds, which are the subject ofChapter 7. It was exactly this particular feature of Gorenstein simplicial posetswhich allowed Masuda [207] to complete the characterisation of their h-vectors,conjectured by Stanley in [292]. We include Masuda’s result here as Theorem 3.7.4.

Definition 3.7.1. A simplicial poset S is Gorenstein (respectively, Goren-stein* ) if its barycentric subdivision S ′ is a Gorenstein (respectively, Gorenstein*)simplicial complex.

Like Cohen–Macaulayness, the property of a simplicial poset S being Goren-stein* depends only on the topology of the realisation |S| (this follows from Theo-rem 3.4.2). In particular, simplicial cell subdivisions of spheres are Gorenstein*.

The problem of characterisation of h-vectors of Gorenstein* simplicial posetsis more subtle than the corresponding question in the Cohen–Macaulay case. (Al-though this problem is much easier for simplicial posets than for simplicial com-plexes, see the discussion of the g-conjecture in Sections 2.5 and 3.4.)


Theorem 3.7.2. Let h(S) = (h0, h1, . . . , hn) be the h-vector of a Gorenstein*simplicial poset of rank n. Then h0 = 1, hi > 0 and hi = hn−i for any i.

Proof. The inequalities hi > 0 follow from the fact that S is Cohen–Macaulay(Theorem 3.6.8). The identities hi = hn−i will follow from the expression of theh-vector of the barycentric subdivision S ′ via h(S) and from the Dehn–Sommervillerelations for the Gorenstein* simplicial complex S ′. Indeed, repeating the argumentfrom Lemmata 2.3.4 and 2.3.5 we obtain the identity h(S ′) = Dh(S), in which thevector h(S ′) is symmetric, i.e. satisfies the Dehn–Sommerville relations. It can bechecked directly using some identities for binomial coefficients that the operator D(and its inverse) takes symmetric vectors to symmetric ones (which is equivalentto the identity dpq = dn+1−p,n+1−q). This calculation can be avoided by usingthe following argument. The Dehn–Sommerville relations specify a linear subspaceW of dimension k =

[n2

]+ 1 in the space Rn+1 with coordinates h0, . . . , hn. We

need to check that this subspace is D-invariant. To do this it suffices to choosea basis e1, . . . , ek in W and check that De i ∈ W for all i. There is a basis inW consisting of h-vectors of simplicial spheres (and even simplicial polytopes, seethe proof of Proposition 1.4.1). Since the barycentric subdivision of a simplicialsphere is a simplicial sphere, the vectors Dei, 1 6 i 6 k, are also symmetric,and W is a D-invariant subspace. Thus, the vector h(S) = D−1h(S ′) satisfies theDehn–Sommerville relations.

Theorem 3.7.3 ([292, Theorem 4.3]). Let h = (h0, h1, . . . , hn) be an integervector with h0 = 1, hi > 0 and hi = hn−i. Any of the following (mutually exclusive)conditions are sufficient for the existence of a Gorenstein* simplicial poset of rankn and h-vector h(S) = h:

(a) n is odd;(b) n is even and hn/2 is even;(c) n is even, hn/2 is odd, and hi > 0 for all i.

Proof. We start with the following two basic examples of (n− 1)-dimensionalsimplicial cell complexes of dimension: ∂∆n, with h-vector h(∂∆n) = (1, 1, . . . , 1);and Sn, the simplicial cell complex obtained by identifying two (n − 1)-simplicesalong their boundaries, with h(Sn) = (1, 0, . . . , 0, 1). By applying the standardoperations of join and connected sum (Constructions 2.2.8 and 2.2.11) to these twocomplexes we can obtain a simplicial cell complex with any prescribed h-vectorsatisfying the conditions of the theorem. Indeed, for k 6= n− k we have

h(Sk ∗ Sn−k) = (1, 0, . . . , 0, 1, 0, . . . , 0, 1, 0, . . . , 0, 1),

where hk = hn−k = 1, and the other entries are zero. Also, for n = 2k we have

h(Sk ∗ Sk) = (1, 0, . . . , 0, 2, 0, . . . , 0, 1),

where hk = 2. Now, by taking connected sum of the appropriate number of com-plexes ∂∆n, Sn and Sk ∗ Sn−k and using the identity

hi(S # S) = hi(S) + hi(S) for 1 6 i 6 n− 1,

(see Example 1.3.8, which is valid for any two pure (n− 1)-dimensional simplicialcell complexes), we obtain any required h-vector.

The subtlest part of the characterisation of h-vectors of Gorenstein* simplicialposets is the following result, which was proved by Masuda:

3.8. GENERALISED DEHN–SOMMERVILLE RELATIONS 133

Theorem 3.7.4 ([207]). Let h(S) = (h0, h1, . . . , hn) be the h-vector of a Goren-stein* simplicial poset S of even rank n, and let hi = 0 for some i. Then the numberhn/2 is even.

Note that the evenness of hn/2 is equivalent to the evenness of the number

of facets fn−1 =∑ni=0 hi. The idea behind Masuda’s proof of Theorem 3.7.4 lies

within the topological theory of torus manifolds, which is the subject of Section 7.4.We combine the results of Theorems 3.7.2, 3.7.3 and 3.7.4 in the following

characterisation result for the h-vectors of Gorenstein* simplicial posets.

Theorem 3.7.5. An integer vector h = (h0, h1, . . . , hn) is the h-vector of aGorenstein* simplicial poset of rank n (or a simplicial cell subdivision of an (n−1)-sphere) if and only if the following conditions are satisfied:

(a) h0 = 1 and hi > 0;(b) hi = hn−i for all i;(c) either hi > 0 for all i or

∑ni=0 hi is even.

3.8. Generalised Dehn–Sommerville relations

In this section we obtain some further generalisations of the Dehn–Sommervillerelations, in particular, to arbitrary triangulated manifolds.

Let S be a simplicial poset of rank n. Given σ ∈ S, consider the closed uppersemi-interval S>σ = τ ∈ S : τ > σ with the induced rank function, and set

(3.20) χ(S>σ) =∑

τ>σ

(−1)|τ |−1.

A simplicial poset S of rank n satisfying χ(S>σ) = (−1)n−1 for all σ ∈ Sis called Eulerian. According to a result of [290, (3.40)], the Dehn–Sommervillerelations hi = hn−i hold for Eulerian posets. This can be generalised as follows.

Theorem 3.8.1 (see [204, Theorem 9.1]). The following identity holds for theh-vector h(S) = (h0, . . . , hn) of a simplicial poset S of rank n:

n∑

i=0

(hn−i − hi)ti =∑

σ∈S

(1 + (−1)nχ(S>σ)

)(t− 1)n−|σ|.

In particular, if S is Eulerian, then hi = hn−i.

Proof. We haven∑

i=0

hiti = tn

n∑

i=0

hi(1t )n−i = tn

n∑

i=0

fi−1

(1−tt

)n−i

=

n∑

i=0

fi−1ti(1− t)n−i =

∑

τ∈S

t|τ |(1− t)n−|τ |

=∑

τ∈S

∑

σ6τ

(t− 1)|τ |−|σ|(1− t)n−|τ | =∑

τ∈S

∑

σ6τ

(−1)n−|τ |(t− 1)n−|σ|

=∑

σ∈S

(t− 1)n−|σ|∑

τ>σ

(−1)n−|τ | =∑

σ∈S

(t− 1)n−|σ|(−1)n−1χ(S>σ),

(3.21)

where the fifth identity follows from the binomial expansion of the right hand sideof the identity t|τ | = ((t − 1) + 1)|τ | and the fact that [0, τ ] = σ ∈ S : σ 6 τ is aBoolean lattice of rank |τ |.


On the other hand, we have

(3.22)

n∑

i=0

hn−iti =

n∑

i=0

hitn−i =

n∑

i=0

fi−1(t− 1)n−i =∑

σ∈S

(t− 1)n−|σ|.

Subtracting (3.21) from (3.22) we obtain the required identity.

As a corollary we obtain a generalisation of the Dehn–Sommerville relationsto triangulated manifolds. This formula appeared in [293, p. 74] (the orientabilityassumption there can be removed by passing to the orientation double cover, seealso [54, Corollary 4.5.4]):

Theorem 3.8.2. Let K be a triangulation of a closed (n−1)-dimensional mani-fold. Then the h-vector h(K) = (h0, . . . , hn) satisfies the identities

hn−i − hi = (−1)i(n

i

)(χ(K)− χ(Sn−1)

), 0 6 i 6 n.

Here χ(K) = f0−f1+· · ·+(−1)n−1fn−1 = 1+(−1)n−1hn is the Euler characteristicof K and χ(Sn−1) = 1 + (−1)n−1.

Proof. Viewing K as a simplicial poset, we calculate

χ(K>σ) =∑

τ>σ

(−1)|τ |−1 + (−1)|σ|−1 = (−1)|σ|(∑

τ>σ

(−1)|τ |−|σ|−1 − 1)

= (−1)|σ|( ∑

∅ 6=ρ∈lkK σ

(−1)|ρ|−1 − 1)= (−1)|σ|

(χ(lkK σ) − 1

).

Here we have used the fact that the poset of nonempty faces of lkK σ is isomorphicto S>σ, with the rank function shifted by |σ|. Now since K is a triangulated (n−1)-dimensional manifold, the link of a nonempty face σ ∈ K has the homology of asphere of dimension (n−|σ|−1). Hence, χ(lkK σ) = 1+(−1)n−|σ|−1, and thereforeχ(K>σ) = (−1)n−1 for σ 6= ∅. Also, lkK ∅ = K. Now using the identity ofTheorem 3.8.1 we calculaten∑

i=0

(hn−i−hi)ti =(1+(−1)n(χ(K)−1)

)(t−1)n = (−1)n

(χ(K)−χ(Sn−1)

)(t−1)n.

The required identity follows by comparing the coefficients of ti.

For other generalisations of Dehn–Sommerville relations see [27] and [138].

Exercises.

3.8.3. The identity of Theorem 3.8.2 holds for simplicial posets.

CHAPTER 4

Moment-angle complexes

This is the first genuinely ‘toric’ chapter of this book; it links the combinatorialand algebraic constructions of the previous chapters to the world of toric spaces.

The term ‘moment-angle complex’ refers to a decomposition of a certain toricspace ZK into products of polydiscs and tori parametrised by simplices in a givensimplicial complex K. The underlying space ZK features in several important al-gebraic, symplectic and topological constructions of torus actions, which are thesubject of the next three chapters. The decomposition of ZK as the ‘moment-angle complex’, which first appeared in [52], [53], provided an effective topologicalinstrument for studying these spaces.

The basic building block in the ‘moment-angle’ decomposition of ZK is thepair (D2, S1) of a unit disc and circle, and the whole construction can be extendednaturally to arbitrary pairs of spaces (X,A). The resulting complex (X,A)K isnow known as the ‘polyhedral product space’ over a simplicial complex K; thisterminology was suggested by William Browder, cf. [14]. Many spaces importantfor toric topology admit polyhedral product decompositions.

It has soon become clear that the construction of the moment-angle complexZK and its generalisation (X,A)K is of truly universal nature, and has remarkablefunctorial properties. The most basic of these is that the construction of ZK es-tablishes a functor from simplicial complexes and simplicial maps to spaces withtorus actions and equivariant maps. If K is a simplicial subdivision of a sphere (atriangulated sphere), then ZK is a manifold, and most important geometric exam-ples of ZK arise in this way. In the case when A is a point, the polyhedral product(X, pt)K interpolates between the m-fold wedge, or bouquet, of X (correspondingto m discrete points as K) and the m-fold product of X (corresponding to the fullsimplex as K). Parallel to the topological and geometric study of moment-anglecomplexes ZK and related toric spaces, a homotopy-theoretic study of polyhedralproducts (X,A)K has now gained its own momentum. Basic homotopy properties ofmoment-angle complexes are given in Section 4.3, while more advanced homotopy-theoretic aspects of toric topology are the subject of Chapter 8.

The key result of this chapter is the calculation of the integral cohomology ringof ZK, carried out in Section 4.5. The ring H∗(ZK) is shown to be isomorphic to theTor-algebra TorZ[v1,...,vm](Z[K],Z), where Z[K] is the face ring of K. The canonicalbigraded structure in the Tor groups thereby acquires a geometric interpretationin terms of the bigraded cell decomposition of ZK. The calculation of H∗(ZK)builds upon a construction of a ring model for cellular cochains of ZK and thecorresponding cellular diagonal approximation, which is functorial with respect tomaps of moment-angle complexes induced by simplicial maps of K. This functorialproperty of the cellular diagonal approximation for ZK is quite special, due to thelack of such a construction for general cell complexes.

135

136 4. MOMENT-ANGLE COMPLEXES

The construction of the moment-angle complex therefore brings the methods ofequivariant topology to bear on the study of combinatorics of simplicial complexes,and gives a new geometric dimension to combinatorial commutative algebra. Inparticular, homological invariants of face rings, such as Tor-algebras or algebraicBetti numbers, can be now interpreted geometrically in terms of cohomology ofmoment-angle complexes. This link is explored further in Sections 4.6 and 4.10.

Another important aspect of the theory of moment-angle complexes is theirconnection to coordinate subspace arrangements and their complements. As wehave already seen in Proposition 3.1.12, coordinate subspace arrangements ariseas affine varieties corresponding to face rings. Their complements have played animportant role in toric geometry and singularity theory, and, more recently, in thetheory of linkages and robotic motion planning. Arrangements of coordinate sub-spaces in Cm correspond bijectively to simplicial complexes K on the set [m], andthe complement of such an arrangement deformation retracts onto the correspond-ing moment-angle complex ZK. In particular, the moment-angle complex and thecomplement of the arrangement have the same homotopy type. We therefore mayuse the results on moment-angle complexes to obtain a description of the coho-mology groups and cup product structure of a coordinate subspace arrangementcomplement. The formula obtained for the cohomology groups is related to thegeneral formula of Goresky–MacPherson [130] by means of Alexander duality.

The material of this chapter, with the exception of ‘Additional topics’, is mainlya freshened and modernised exposition of the results obtained by the authors andtheir collaborators in [53], [55], [24], [257].

Spaces with torus actions, or toric spaces , will be the main players throughoutthe rest of this book. (See Appendix B.3 for the key concepts of the theory ofgroup actions on topological spaces.) The most basic example of a toric space isthe complex m-dimensional space Cm, on which the standard torus

Tm =t = (t1, . . . , tm) ∈ Cm : |ti| = 1 for i = 1, . . . ,m

acts coordinatewise. That is, the action is given by

Tm × Cm −→ Cm,

(t1, . . . , tm) · (z1, . . . , zm) = (t1z1, . . . , tmzm).

The quotient Cm/Tm of this action is the positive orthant

Rm> =(y1, . . . , ym) ∈ Rm : yi > 0 for i = 1, . . . ,m

,

with the quotient projection given by

µ : Cm −→ Rm> ,

(z1, . . . , zm) 7−→ (|z1|2, . . . , |zm|2)

(or by (z1, . . . , zm) 7−→ (|z1|, . . . , |zm|), but the former is usually more preferable).We shall use the blackboard bold capital T in the notation for the standard torus

Tm only, and use italic Tm to denote an abstract m-torus, i.e. a compact abelianLie group isomorphic to a product of m circles. We shall also denote the standardunit circle by S or T occasionally, to distinguish it from an abstract circle S1.

All homology and cohomology groups in this chapter are with integer coeffi-cients, unless another coefficient group is explicitly specified.

4.1. BASIC DEFINITIONS 137

4.1. Basic definitions

Moment-angle complex ZK. We consider the unit polydisc in the m-dimensional complex space Cm:

Dm =(z1, . . . , zm) ∈ Cm : |zi|2 6 1 for i = 1, . . . ,m

.

The polydisc Dm is a Tm-invariant subspace of Cm, and the quotient Dm/Tm isidentified with the standard unit cube Im ⊂ Rm> .

Construction 4.1.1 (moment-angle complex). Let K be a simplicial com-plex on the set [m]. We recall the cubical subcomplex cc(K) in Im from Construc-tion 2.9.11, which subdivides coneK. The moment-angle complex ZK correspondingto K is defined from the pullback square

ZK

//

Dm

µ

cc(K) // Im

Explicitly, ZK = µ−1(cc(K)). By construction, ZK is a Tm-invariant subspace inthe polydisc Dm, and the quotient ZK/Tm is homeomorphic to | coneK|.

Using the decomposition cc(K) = ⋃I∈KCI into faces, see (2.10), it follows that

(4.1) ZK =⋃

I∈K

BI ,

where

BI = µ−1(CI) =(z1, . . . , zm) ∈ Dm : |zj |2 = 1 for j /∈ I

,

and the union in (4.1) is understood as the union of subsets inside the polydisc Dm.Note that BI is a product of |I| discs and m− |I| circles. Following our notationaltradition, we denote a topological 2-disc (the underlying space of D) by D2. Thenwe may rewrite (4.1) as the following decomposition of ZK into products of discsand circles:

(4.2) ZK =⋃

I∈K

(∏

i∈I

D2 ×∏

i/∈I

S1),

From now on we shall denote the space BI by (D2, S1)I . Obviously, the unionin (4.1) or (4.2) can be taken over the maximal simplices I ∈ K only.

Using the categorical language, we may consider the face category cat(K), anddefine the functor (or diagram, see Appendix C.1)

(4.3)DK(D

2, S1) : cat(K) −→ top,

I 7−→ BI = (D2, S1)I ,

which maps the morphism I ⊂ J of cat(K) to the inclusion of spaces (D2, S1)I ⊂(D2, S1)J . Then we have

ZK = colimDK(D2, S1) = colim

I∈K(D2, S1)I .

Example 4.1.2.1. Let K = ∆m−1 be the full simplex. Then cc(K) = Im and ZK = Dm.


2. Let K be a simplicial complex on [m], and let K be the complex on [m+1]obtained by adding one ghost vertex = m+ 1 to K. Then the cubical complexcc(K) is contained in the facet ym+1 = 1 of the cube Im+1, and

ZK = ZK × S1.

In particular, if K is the ‘empty’ simplicial complex on [m], consisting of the emptysimplex ∅ only, then cc(K) is the vertex (1, . . . , 1) ∈ Im and ZK = µ−1(1, . . . , 1) =Tm is the standard m-torus.

For arbitrary K on [m], the moment-angle complex ZK contains the m-torusTm (corresponding to K = ∅) and is contained in the polydisc Dm (correspondingto K = ∆m−1).

3. Let K be the complex consisting of two disjoint points. Then

ZK = (D2 × S1) ∪ (S1 ×D2) = ∂(D2 ×D2) ∼= S3,

the standard decomposition of a 3-sphere into the union of two solid tori.4. More generally, if K = ∂∆m−1 (the boundary of a simplex), then

ZK = (D2 × · · · ×D2 × S1) ∪ (D2 × · · · × S1 ×D2) ∪ · · · ∪ (S1 × · · · ×D2 ×D2)

= ∂((D2)m

) ∼= S2m−1.

5. Let K = rr rr1 3

4 2

, the boundary of a 4-gon. Then we have four maximalsimplices 1, 3, 2, 3, 1, 4 and 2, 4, and

ZK = (D2 × S1 ×D2 × S1) ∪ (S1 ×D2 ×D2 × S1)

∪ (D2 × S1 × S1 ×D2) ∪ (S1 ×D2 × S1 ×D2)

=((D2 × S1) ∪ (S1 ×D2)

)×D2 × S1 ∪

((D2 × S1) ∪ (S1 ×D2)

)× S1 ×D2

=((D2 × S1) ∪ (S1 ×D2)

)×((D2 × S1) ∪ (S1 ×D2)

) ∼= S3 × S3.

In the last example, K is the join of 1, 2 and 3, 4. More generally,

Proposition 4.1.3. Let K = K1 ∗ K2; then

ZK = ZK1 ×ZK2 .

Proof. By the definition of join (Construction 2.2.8), we have

ZK1∗K2 =⋃

I1∈K1, I2∈K2

(D2, S1)I1⊔I2 =⋃

I1∈K1, I2∈K2

(D2, S1)I1 × (D2, S1)I2

=( ⋃

I1∈K1

(D2, S1)I1)×( ⋃

I2∈K2

(D2, S1)I2)= ZK1 ×ZK2 .

The topological structure of ZK is quite complicated even for simplicial com-plexes K with few vertices. Several different techniques will be developed to describethe topology of ZK; this is one of the main subjects of the book. To get an idea onhow ZK may look like we included Exercises 4.2.9 and 4.2.10, in which the topo-logical structure of ZK is more complicated than in the examples above, but whichare still accessible by relatively elementary topological methods. As we shall seebelow, the cohomology of ZK may have an arbitrary torsion (Corollary 4.5.9), aswell as Massey products (Section 4.9).

Moment-angle complexes corresponding to triangulated spheres and manifoldsare of particular interest:

4.1. BASIC DEFINITIONS 139

Theorem 4.1.4. Let K be a triangulation of an (n−1)-dimensional sphere withm vertices. Then ZK is a (closed) topological manifold of dimension m+ n.

If K is a triangulated manifold, then ZK \ µ−1(1, . . . , 1) is an (open) non-compact manifold. Here (1, . . . , 1) ∈ Im is the cone vertex, and µ−1(1, . . . , 1) = Tm.

Proof. We first construct a decomposition of the polyhedron | cone(K′)| into‘faces’ similar to the faces of a simple polytope (in fact, in the case of the nervecomplex K = KP , our faces will be exactly the faces of P ). The vertices i ∈ K arealso vertices of the barycentric subdivision K′, and we set

(4.4) Fi = stK′i, 1 6 i 6 m,

(i.e. Fi is the star of the ith vertex of K in the barycentric subdivision K′). Werefer to Fi as facets of our face decomposition, and define a face of codimension kas a nonempty intersection of a set of k facets. In particular, the vertices of our facedecomposition are the barycentres of (n − 1)-dimensional simplices of K. Such abarycentre b corresponds to a maximal simplex I = i1, . . . , in of K, and we denoteby UI the open subset in | cone(K′)| obtained by removing all faces not containing b.We observe that any point of | coneK′| is contained in UI for some I ∈ K.

Under the map cone(ic) : | cone(K′)| → Im of Construction 2.9.11 the facetFi is mapped to the intersection of cc(K) with the ith coordinate plane yi = 0.Therefore, the image of UI under the map cone(ic) is given by

(4.5) WI = cone(ic)(UI) = (y1, . . . , ym) ∈ cc(K) : yi 6= 0 for i /∈ I.Assume now that K is a triangulated sphere. Then | coneK′| is homeomorphic

to an n-dimensional disc Dn, and each UI is homeomorphic to an open subset inRn> preserving the dimension of faces. (This means that | coneK′| is a manifold

with corners , see Section 7.1). By identifying | coneK′| with cc(K) and furtheridentifying cc(K) with the quotient ZK/Tm, we obtain that each point of ZK hasa neighbourhood of the form µ−1(WI). It follows from (4.5) that the latter ishomeomorphic to an open subset in µ−1

n (Rn>)× Tm−n = Cn × Tm−n, where Rn isthe coordinate n-plane corresponding to i1, . . . , in, the map µn : Cn → Rn> is therestriction of µ : Cm → Rm> to the corresponding coordinate plane in Cm, and the

torus Tm−n sits in the complementary coordinate (m − n)-plane in Cm. An opensubset in Cn × Tm−n with n > 1 can be regarded as an open subset in Rm+n, andtherefore ZK is an (m+ n)-dimensional manifold.

If K is a triangulated manifold, then | coneK′| is not a manifold because of thesingularity at the cone vertex v. However, by removing this vertex we obtain a (non-compact) manifold whose boundary is |K|. Using the face decomposition definedby (4.4) we obtain that | coneK′| \ v is locally homeomorphic to Rn> preserving

the dimension of faces (i.e. | coneK′| \ v is a non-compact manifold with corners).Under the identification of | coneK′| with cc(K) the vertex of the cone is mappedto the vertex (1, . . . , 1) ∈ Im, and µ−1(1, . . . , 1) = Tm. Therefore,

µ−1(cc(K)\(1, . . . , 1)

)= ZK \ µ−1(1, . . . , 1)

is an (m+ n)-dimensional non-compact manifold.

Remark. A pair of spaces (X,A) where A is a compact subset in X is calleda Lefschetz pair if X \ A is an open (non-compact) manifold. We therefore obtainthat (ZK, µ

−1(1, . . . , 1)) is a Lefschetz pair whenever K is a triangulated manifold.


We therefore refer to moment-angle complexes ZK corresponding to triangu-lated spheres as moment-angle manifolds. Polytopal moment-angle manifolds ZKP

,corresponding to the nerve complexes KP of simple polytopes (see Example 2.2.4),are particularly important. As we shall see in Chapter 6, polytopal moment-anglemanifolds are smooth. A smooth structure also exists on moment-angle manifoldsZK corresponding to starshaped spheres K (i.e. underlying complexes of completesimplicial fans, see Section 2.5). In general, the smoothness of ZK is open.

The geometry of moment-angle manifolds is nice and rich; it is the subject ofChapter 6.

Real moment-angle complex RK. The construction of the moment-anglecomplex ZK has a real analogue, in which the complex space Cm is replaced by thereal space Rm, the complex polydisc Dm is replaced by the ‘big’ cube

[−1, 1]m =(u1, . . . , um) ∈ Rm : |ui|2 6 1 for i = 1, . . . ,m

,

the standard torus Tm is replaced by the ‘real torus’ Zm2 (the product of m copiesof the group Z2 = −1, 1), and the pair (D2, S1) is replaced by (D1, S0), whereS0 (a pair of points) is the boundary of the segment D1. The group (Z2)

m acts onthe big cube [−1, 1]m coordinatewise, with quotient the standard ‘small’ cube Im.The quotient projection [−1, 1]m → Im may be described by the map

ρ : (u1, . . . , um) 7→ (u21, . . . , u2m).

Construction 4.1.5 (real moment-angle complex). Given a simplicial com-plex K on [m], define the real moment-angle complex RK from the pullback square

RK

//

[−1, 1]m

ρ

cc(K) // Im

Explicitly, RK = ρ−1(cc(K)). By construction, RK is a Zm2 -invariant subspace inthe ‘big’ cube [−1, 1]m, and the quotient RK/Zm2 is homeomorphic to | coneK|.RK is a cubical subcomplex in [−1, 1]m obtained by reflecting the subcomplex

cc(K) ⊂ Im = [0, 1]m at all m coordinate hyperplanes of Rm. If K = KP is thenerve complex of a simple polytope P , then cc(KP ) can be viewed as a cubicalsubdivision of P embedded piecewise linearly into Rm> (see Construction 2.9.7). Inthis case RKP

is obtained by reflecting the image of P at all coordinate planes.By analogy with (4.2), we have

RK =⋃

I∈K

(∏

i∈I

D1 ×∏

i/∈I

S0).

Example 4.1.6.1. LetK = ∂∆m−1 be the boundary of the standard simplex. ThenRK

∼= Sm−1

is the boundary of the cube [−1, 1]m. For m = 3 this complex is obtained byreflecting the complex shown in Fig. 2.14 (b) at all 3 coordinate planes of R3.

2. Let K consist of m disjoint points. Then RK is the 1-dimensional skeleton(graph) of the cube [−1, 1]m. For m = 3 this complex is obtained by reflecting thecomplex shown in Fig. 2.14 (a) at all 3 coordinate planes of R3.

4.2. POLYHEDRAL PRODUCTS 141

3. More generally, let K = ski∆m−1 be the i-dimensional skeleton of ∆m−1 (i.e.the set of all faces of ∆m−1 of dimension 6 i). Then RK is the (i+ 1)-dimensionalskeleton of the cube [−1, 1]m.

The following analogue of Theorem 4.1.4 holds, and is proved similarly:

Theorem 4.1.7. Let K be a triangulation of an (n−1)-dimensional sphere withm vertices. Then RK is a (closed) topological manifold of dimension n.

If K is a triangulated manifold, then RK \ ρ−1(1, . . . , 1) is an (open) non-compact manifold, where ρ−1(1, . . . , 1) = −1, 1m.

The real moment-angle complexes corresponding to polygons can be identifiedeasily (compare Exercise 4.2.10):

Proposition 4.1.8. Let K be the boundary of an m-gon. Then RK is homeo-morphic to an oriented surface Sg of genus g = 1 + (m− 4)2m−3.

Proof. We observe that the manifold RK is orientable (an exercise). Sinceit is 2-dimensional, its topological type is determined by the Euler characteristic.Now RK is obtained by reflecting the m-gon embedded into Rm> at m coordinatehyperplanes, so that RK is patched from 2m polygons, meeting by 4 at each vertex.Therefore, the number of vertices is m2m−2, the number of edges is m2m−1, andthe Euler characteristic is

χ(RK) = 2m−2(4 −m) = 2− 2g.

The result follows.

4.2. Polyhedral products

Decomposition (4.2) of ZK which uses the disc and circle (D2, S1) is readilygeneralised to arbitrary pairs of spaces:

Construction 4.2.1 (polyhedral product). Let K be a simplicial complexon [m] and

(X ,A) = (X1, A1), . . . , (Xm, Am)be a collection of m pairs of spaces, Ai ⊂ Xi. For each simplex I ∈ [m] we set

(4.6) (X ,A)I =(x1, . . . , xm) ∈

m∏

j=1

Xj : xj ∈ Aj for j /∈ I

and define the polyhedral product of (X ,A) corresponding to K by

(X ,A)K =⋃

I∈K

(X ,A)I =⋃

I∈K

(∏

i∈I

Xi ×∏

i/∈I

Ai

).

Using the categorical language, we can define the cat(K)-diagram

(4.7)DK(X ,A) : cat(K) −→ top,

I 7−→ (X ,A)I ,

which maps the morphism I ⊂ J of cat(K) to the inclusion of spaces (X ,A)I ⊂(X ,A)J . Then we have

(X ,A)K = colimDK(X ,A) = colimI∈K

(X ,A)I .


In the case when all the pairs (Xi, Ai) are the same, i.e. Xi = X and Ai = Afor i = 1, . . . ,m, we use the notation (X,A)K for (X ,A)K. Also, if each Xi is a

pointed space and Ai = pt , then we use the abbreviated notation XK for (X , pt)K,and XK for (X, pt)K.

Remark. The decomposition of ZK into a union of products of discs andcircles first appeared in [52] (in the polytopal case) and in [54] (in general). Theterm ‘moment-angle complex’ for ZK = (D2, S1)K was also introduced in [54],where several other examples of polyhedral products (X,A)K were considered. Thedefinition of (X,A)K for an arbitrary pair of spaces (X,A) was suggested to theauthors by N. Strickland (in a private communication, and also in an unpublishednote) as a general framework for the constructions of [54]; it was also included inthe final version of [54] and in [55]. Further generalisations of (X,A)K to a setof pairs of spaces (X ,A) were studied in the work of Grbic and Theriault [133],as well as Bahri, Bendersky, Cohen and Gitler [14], where the term ‘polyhedralproduct’ was introduced (following a suggestion of W. Browder). Since 2000, theterms ‘generalised moment-angle complex’, ‘K-product’ and ‘partial product space’have been also used to refer to the spaces (X,A)K.

Recall that a map of pairs (X,A)→ (X ′, A′) is a commutative diagram

(4.8)A → X↓ ↓A′ → X ′.

We refer to (X,A) as a monoid pair if X is a topological monoid (a space with acontinuous associative multiplication and unit), and A is a submonoid. A map ofmonoid pairs is a map of pairs in which all maps in (4.8) are homomorphisms.

If (X,A) is a monoid pair, then a set map ϕ : [l]→ [m] induces a map

(4.9) ψ :

l∏

i=1

X →m∏

i=1

X, (x1, . . . , xl) 7→ (y1, . . . , ym),

whereyj =

∏

i∈ϕ−1(j)

xi, for j = 1, . . . ,m,

and we set yj = 1 if ϕ−1(j) = ∅.

Proposition 4.2.2. If (X,A) is a monoid pair, then (X,A)K is an invariantsubspace of

∏mi=1X with respect to the coordinatewise action of

∏mi=1 A on

∏mi=1X.

Proof. Indeed, (X,A)I ⊂∏mi=1X is an invariant subset for each I ∈ K.

The following proposition describes the functorial properties of the polyhedralproduct in its K and (X ,A) arguments.

Proposition 4.2.3.

(a) A set of maps of pairs (X,A) → (X′,A′) induces a map of polyhedralproducts (X,A)K → (X′,A′)K. If two sets of maps (X,A)→ (X′,A′) arecomponentwise homotopic, then the induced maps (X,A)K → (X′,A′)K

are also homotopic.(b) An inclusion of a simplicial subcomplex L → K induces an inclusion of

polyhedral products (X,A)L → (X,A)K.

4.2. POLYHEDRAL PRODUCTS 143

(c) If (X,A) is a monoid pair, then for any simplicial map ϕ : L → K ofsimplicial complexes on the sets [l] and [m] respectively, the map (4.9)restricts to a map of polyhedral products ϕZ : (X,A)L → (X,A)K.

(d) If (X,A) is a commutative monoid pair, then the restriction

ψ|A :

l∏

i=1

A→m∏

i=1

A

of (4.9) is a homomorphism, and the induced map ϕZ : (X,A)L →(X,A)K is ψ|A-equivariant, i.e.

ϕZ(a · x) = ψ|A(a) · ϕZ(x)

for all a = (a1, . . . , al) ∈∏li=1A and x = (x1, . . . , xl) ∈ (X,A)L.

Proof. For (a), we observe that a set of maps (X ,A) → (X ′,A′) induces amap (X ,A)I → (X ′,A′)I for each I ∈ K, and these maps corresponding to differ-ent I, J ∈ K are compatible on the intersection (X ,A)I ∩ (X ,A)J = (X ,A)I∩J .We therefore obtain a map (X ,A)K → (X ′,A′)K. A componentwise homotopybetween two maps (X ,A) → (X ′,A′) can be thought of as a map of pairs(X × I,A × I) → (X ′,A′), where X × I consists of spaces Xi × I. It thereforeinduces a map of polyhedral products

(X × I,A× I)K → (X ′,A′)K

where (X × I,A × I)K ∼= (X ,A)K × (I, I)K = (X ,A)K × Im. By restricting theresulting map (X ,A)K× Im → (X ′,A′)K to the diagonal of the cube Im we obtaina homotopy between the two induced maps (X ,A)K → (X ′,A′)K.

To prove (b) we just observe that if L is a subcomplex of K, then for each I ∈ Lwe have (X ,A)I ⊂ ZK.

To prove (c) we observe that for any subset I ⊂ [m] we have ψ((X,A)I

)⊂

(X,A)ϕ(I). Let I ∈ L, so that (X,A)I ⊂ (X,A)L. Since ϕ is a simplicial map, wehave ϕ(I) ∈ K and (X,A)ϕ(I) ⊂ (X,A)K. Therefore, the map ψ restricts to a mapof polyhedral products (X,A)L → (X,A)K.

Statement (d) is proved by a direct calculation:

ϕZ(a · x ) = ϕZ(a1x1, . . . , alxl) =( ∏

i∈ϕ−1(1)

aixi, . . . ,∏

i∈ϕ−1(m)

aixi

)

=( ∏

i∈ϕ−1(1)

ai∏

i∈ϕ−1(1)

xi, . . . ,∏

i∈ϕ−1(m)

ai∏

i∈ϕ−1(m)

xi

)

=( ∏

i∈ϕ−1(1)

ai, . . . ,∏

i∈ϕ−1(m)

ai

)·( ∏

i∈ϕ−1(1)

xi, . . . ,∏

i∈ϕ−1(m)

xi

)

= ψ|A(a) · ϕZ(x ).

Note that we have used the commutativity of X in the third identity.

We state the most important particular case of Proposition 4.2.3 separately:

Proposition 4.2.4. A simplicial map L → K of simplicial complexes on thesets [l] and [m] gives rise to a map of moment-angle complexes ZL → ZK which isequivariant with respect to the induced homomorphism of tori Tl → Tm.


The polyhedral product construction behaves nicely with respect to the joinoperation (the proof is exactly the same as that of Proposition 4.1.3):

Proposition 4.2.5. We have that

(X,A)K1∗K2 = (X,A)K1 × (X,A)K2 .

Example 4.2.6.1. The moment-angle complex ZK is the polyhedral product (D2, S1)K (when

considered abstractly) or (D, S)K (when viewed as a subcomplex in Dm).2. For the cubical complex of (2.13) we have

cc(K) = (I1, 1)K,

where I1 = [0, 1] is the unit interval and 1 is its edge. The quotient map ZK →| coneK| is the map of polyhedral products (D2, S1)K → (I1, 1)K induced by themap of pairs (D2, S1)→ (I1, 1), which is the quotient map by the S1-action.

3. For the real moment-angle complex we have

RK = (D1, S0)K,

Where D1 = [−1, 1] is a 1-disc, and S0 = −1, 1 is its boundary.4. If K consists of m disjoint points and Ai = pt , then

(X , pt)K = XK = X1 ∨X2 ∨ · · · ∨Xm

is the wedge (or bouquet) of the Xi’s.5. More generally, consider the sequence of inclusions of skeleta

sk0∆m−1 ⊂ · · · ⊂ skm−2∆m−1 ⊂ skm−1∆m−1 = ∆m−1.

It gives rise to a filtration in the product X1 ×X2 × · · · ×Xm:

X sk0∆m−1 ⊂ · · · ⊂ X skm−2∆m−1 ⊂ X skm−1∆m−1

‖ ‖ ‖X1 ∨X2 ∨ · · · ∨Xm X ∂∆m−1

X1 ×X2 × · · · ×Xm

Its second-to-last term, X ∂∆m−1

is known to topologists as the fat wedge of theXi’s. Explicitly, the fat wedge of a sequence of pointed spaces X1, . . . , Xm is

(X1 × · · · ×Xm−1 × pt) ∪ (X1 × · · · × pt ×Xm) ∪ · · · ∪ (pt × · · · ×Xm−1 ×Xm),

where the union is taken inside the product X1 ×X2 × · · · ×Xm.The filtration above was considered by G. Porter [264], who obtained a decom-

position of its loop spaces into a wedge in the case when each Xi is a suspension,generalising the Hilton–Milnor Theorem. We shall consider this decomposition inmore detail in Section 8.3.

Exercises.

4.2.7. Show that if K is a triangulated sphere, then the manifold RK is ori-entable. (Hint: use the fact that RK is obtained by reflecting the n-ball | coneK|at all m coordinate hyperplanes of Rm to extend the orientation from | coneK| tothe whole of RK.)

4.2.8. Show that if K = ski∆m−1, then RK is homotopy equivalent to a wedgeof (i+1)-dimensional spheres (see Example 4.1.6.3). The number of spheres is given

by∑m

k=i+2

(mk

)(k−1i+1

).

4.3. HOMOTOPICAL PROPERTIES 145

4.2.9. Let K be the complex consisting of three disjoint points. Show that ZK

is homotopy equivalent to the following wedge (bouquet) of spheres:

ZK∼= S3 ∨ S3 ∨ S3 ∨ S4 ∨ S4.

(Hint: compare the case m = 3, i = 0 of the previous exercise; it may also help tolook at the realisation of ZK as the complement of a coordinate subspace arrange-ment (up to homotopy), see Section 4.7.)

4.2.10. Let K be the boundary of a 5-gon. Show that the manifold ZK ishomeomorphic to (S3 × S4)#5, a connected sum of 5 copies of S3 × S4. (This maybe a difficult one; the general statement is given in Theorem 4.6.12 below. Thereader may return to this exercise after reading Section 6.2, see Exercise 6.2.15.)

4.2.11. If K is a triangulation of an (n−1)-sphere with m vertices, then for anyk > 0 the polyhedral product (Dk, Sk−1)K is a manifold of dimension m(k−1)+n.

4.2.12. More generally, if (M,∂M) is a manifold with boundary and K is atriangulated sphere, then (M,∂M)K is a manifold (without boundary). If (M,∂M)is a PL manifold, and K is a PL sphere, then the manifold (M,∂M)K is also PL.

4.2.13. Let KI be the full subcomplex corresponding to I ⊂ [m]. Then ZKIis

a retract of ZK.

4.3. Homotopical properties

Two key observations of this section constitute the basis for the subsequentapplications of the commutative algebra apparatus of Chapter 3 to toric topology.First, the cohomology of the polyhedral product space of the form (CP∞)K =(CP∞, pt)K is isomorphic to the face ring Z[K] (Proposition 4.3.1). Second, themoment-angle complex ZK = (D2, S1)K is the homotopy fibre of the canonicalinclusion (CP∞)K → (CP∞,CP∞)K = (CP∞)m (Theorem 4.3.2).

The classifying space BS1 of the circle S1 is the infinite-dimensional complexprojective space CP∞. The classifying space BTm of them-torus Tm is the product(CP∞)m ofm copies of CP∞. The universal principal S1-bundle is the infinite Hopfbundle S∞ → CP∞ (the direct limit of Hopf bundles S2k+1 → CP k), so the totalspace ETm of the universal principal Tm-bundle over BTm can be identified withthe m-fold product of the infinite-dimensional sphere S∞.

The integral cohomology ring of BTm is isomorphic to the polynomial ringZ[v1, . . . , vm], deg vi = 2 (this explains our choice of grading). The space BTm hasthe canonical cell decomposition, in which each factor CP∞ has one cell in everyeven dimension. The polyhedral product

(CP∞)K =⋃

I∈K

(CP∞, pt)I

is a cellular subcomplex in BTm = (CP∞)m.

Proposition 4.3.1. The cohomology ring of (CP∞)K is isomorphic to the facering Z[K]. The inclusion of a cellular subcomplex

i : (CP∞)K → (CP∞)m

induces the quotient projection in cohomology:

i∗ : Z[v1, . . . , vm]→ Z[v1, . . . , vm]/IK = Z[K].


Proof. Since (CP∞)m has only even-dimensional cells and (CP∞)K is a cel-lular subcomplex, the cohomology of both spaces coincides with their cellularcochains. Let D2k

j denote the 2k-dimensional cell in the jth factor of (CP∞)m.

The cellular cochain group C∗((CP∞)m) has basis of cochains(D2k1j1· · ·D2kp

jp

)∗

dual to the products of cells D2k1j1× · · · ×D2kp

jp. The cochain map

C∗((CP∞)m

)→ C∗

((CP∞)K

)

induced by the inclusion (CP∞)K → (CP∞)m is an epimorphism with kernel

generated by those cochains(D2k1j1· · ·D2kp

jp

)∗for which j1, . . . , jp /∈ K. Under

the identification of C∗((CP∞)m) with Z[v1, . . . , vm], a cochain(D2k1j1· · ·D2kp

jp

)∗is

mapped to the monomial vk1j1 · · · vkpjp

. Therefore, C∗((CP∞)K) is identified with the

quotient of Z[v1, . . . , vm] by the subgroup generated by all monomials vk1j1 · · · vkpjp

with j1, . . . , jp /∈ K. By Proposition 3.1.9, this quotient is exactly Z[K].

We consider the Borel construction ETm ×Tm ZK for the Tm-space ZK (seeAppendix B.3).

Theorem 4.3.2. The inclusion i : (CP∞)K → (CP∞)m is decomposed into acomposition of a homotopy equivalence

h : (CP∞)K≃−→ ETm ×Tm ZK

and the fibration p : ETm ×Tm ZK → BTm = (CP∞)m with fibre ZK.In particular, the moment-angle complex ZK is the homotopy fibre of the canon-

ical inclusion i : (CP∞)K → (CP∞)m.

Proof. We use functoriality and homotopy invariance of the polyhedral prod-uct construction (Proposition 4.2.3). We have ZK =

⋃I∈K(D

2, S1)I , see (4.1), and

each subset (D2, S1)I is Tm-invariant. We therefore obtain the following decompo-sition of the Borel construction as a polyhedral product:

ETm ×Tm ZK =⋃

I∈K

(ETm ×Tm (D2, S1)I

)=⋃

I∈K

(S∞ ×S1 D2, S∞ ×S1 S1

)I

= (S∞ ×S1 D2, S∞ ×S1 S1)K,

where S∞ = ET 1 = ES1.Now consider the commutative diagram

(4.10)

pt −−−−→ S∞ ×S1 S1 −−−−→ pty

yy

CP∞ j−−−−→ S∞ ×S1 D2 f−−−−→ CP∞,

where j is the inclusion of the zero section in a disc bundle (note that CP∞ =S∞/S1), and f is the projection map from the bundle to its Thom space,

f : S∞ ×S1 D2 → (S∞ ×S1 D2)/(S∞ ×S1 S1) ∼= CP∞.

Since S∞×S1 S1 = S∞ and D2 are contractible, the composite maps f j and j fare homotopic to the identity. It follows that we have a homotopy equivalence of

4.3. HOMOTOPICAL PROPERTIES 147

pairs (CP∞, pt)→ (S∞×S1D2, S∞×S1 S1) which induces a homotopy equivalenceof polyhedral products

h : (CP∞)K → (S∞ ×S1 D2, S∞ ×S1 S1)K = ETm ×Tm ZK.

In order to establish the factorisation i = p h we consider the diagram

pt −−−−→ S∞ ×S1 S1 −−−−→ CP∞

yy

∥∥∥

CP∞ j−−−−→ S∞ ×S1 D2 g−−−−→ CP∞,

where g is the projection of the disc bundle onto its base (note that his map isdifferent from the map f above). By passing to the induced maps of polyhedralproducts we obtain the factorisation of

i : (CP∞)Kh−→ (S∞ ×S1 D2, S∞ ×S1 S1)K

p−→ (CP∞,CP∞)K

into the composition of h and p.

The following statement is originally due to Davis and Januszkiewicz [90, Theo-rem 4.8] (they used a different model for ZK, which will be discussed in Section 6.2).

Corollary 4.3.3. The equivariant cohomology ring of the moment-angle com-plex ZK is isomorphic to the face ring of K:

H∗Tm(ZK) ∼= Z[K].

In equivariant cohomology, the projection p : ETm ×Tm ZK → BTm induces thequotient projection

p∗ : Z[v1, . . . , vm]→ Z[K] = Z[v1, . . . , vm]/IK.Proof. This follows from Theorem 4.3.2 and Proposition 4.3.1.

In view of this result, the Borel construction ETm ×Tm ZK is often called theDavis–Januszkiewicz space and denoted by DJ (K). According to Theorem 4.3.2 itis modelled (up to homotopy) on the polyhedral product (CP∞)K.

Example 4.3.4. Let K be the complex consisting of two disjoint points. ThenZK∼= S3 and (CP∞)K = CP∞ ∨ CP∞ (a wedge of two copies of CP∞). The

Borel construction ET 2×T 2 ZK can be identified with the total space of the spherebundle S(η × η) associated with the product of two universal (Hopf) complex linebundles η over BT 1 = CP∞. By Theorem 4.3.2, the space ET 2×T 2ZK is homotopyequivalent to CP∞ ∨ CP∞, and the bundle projection S(η × η) → CP∞ × CP∞

induces the quotient projection Z[v1, v2]→ Z[v1, v2]/(v1v2) in cohomology.

We have the following basic information about the homotopy groups of ZK:

Proposition 4.3.5.

(a) If K is a simplicial complex on the vertex set [m] (i.e. there are noghost vertices), then the moment-angle complex ZK is 2-connected (i.e.π1(ZK) = π2(ZK) = 0), and

πi(ZK) = πi((CP∞)K

)for i > 3.

(b) If K is a q-neighbourly simplicial complex, then πi(ZK) = 0 for i < 2q+1.Furthermore, π2q+1(ZK) is a free abelian group of rank equal to the numberof (q + 1)-element missing faces of K.


Proof. We observe that (CP∞)m is the Eilenberg–Mac Lane space K(Zm, 2),and the 3-dimensional skeleton of (CP∞)K coincides with the 3-skeleton of(CP∞)m. If K is q-neighbourly, then the (2q+1)-skeletons of (CP∞)K and (CP∞)m

agree. Now both statements follow from the exact homotopy sequence of the map(CP∞)K → (CP∞)m with homotopy fibre ZK.

Example 4.3.6. Let K = sk1∆3 (a complete graph on 4 vertices). Then K is

2-neighbourly and has 4 missing triangles, so ZK is 4-connected and π5(ZK) = Z4.

Exercises.

4.3.7. By analogy with Proposition 4.3.1, show that

H∗((RP∞)K,Z2

) ∼= Z2[v1, . . . , vm]/(vi1 · · · vik : i1, . . . , ik /∈ K

), deg vi = 1.

4.3.8. Consider a sequence of pointed odd-dimensional spheres

S = (S2p1−1, . . . , S2pm−1).

Show that there is an isomorphism of rings

H∗((S)K

) ∼= Λ[u1, . . . , um]/(ui1 · · ·uik : i1, . . . , ik /∈ K

), deg ui = 2pi − 1.

This ring is known as the exterior face ring of K. In the case p1 = · · · = pm = 1we obtain (S )K = (S1)K, which is a cell subcomplex in the torus Tm.

For a more general statement describing the cohomology of the polyhedralproduct XK, see Theorem 8.3.7.

4.3.9 ([95, Lemma 2.3.1]). Assume given commutative diagrams

F ′i

//

E′i

//

Bi

Fi // Ei // Bi

of cell complexes where the horizontal arrows are fibrations and the vertical arrowsare inclusions of cell subcomplexes, for i = 1, . . . ,m. Denote by (F ,F ′), (E ,E ′)and (B ,B) the corresponding sequences of pairs. Show that there is a fibration ofpolyhedral products

(F ,F ′)K → (E ,E ′)K → (B ,B)K

where (B ,B)K = B1 × · · · ×Bm.

4.3.10. Use the previous exercise, the path-loop fibration ΩX → PX → X andhomotopy invariance of the polyhedral product (Proposition 4.2.3) to show that

the homotopy fibre of the inclusion XK → Xm is (PX , ΩX )K or, equivalently,(coneΩX , ΩX )K. That is, construct a homotopy fibration

(PX , ΩX )K → (X , pt)K → (X ,X )K.

When Xi = CP∞ we obtain the homotopy fibration ZK → (CP∞)K → (CP∞)m

of Theorem 4.3.2.

4.3.11. When K is a pair of points and X = (X1, X2), show that (PX , ΩX )K

is homotopy equivalent to ΣΩX1∧ΩX2. Deduce Ganea’s Theorem identifying thehomotopy fibre of the inclusion X1 ∨X2 → X1 ×X2 with ΣΩX1 ∧ΩX2.

4.4. CELL DECOMPOSITION 149

4.3.12. In the setting of Example 4.3.4, consider the diagonal circle S1d ⊂ T 2.

Show that it acts freely on ZK∼= S3. Deduce that the Borel construction ET 2×T 2

ZK is homotopy equivalent to ES1 ×S1 CP 1, where CP 1 = S3/S1d with S1-action

given by t · [z0 : z1] = [z0 : tz1]. It follows that ES1×S1 CP 1 ≃ CP∞∨CP∞. Showthat ES1 ×S1 CP 1 can be identified with the complex projectivisation CP (η ⊕ η),where η is the tautological line bundle over CP∞ and η is its complex conjugate.What can be said about the complex projectivisation CP (η ⊕ η)?

4.4. Cell decomposition

We consider the following decomposition of the disc D into 3 cells: the point 1 ∈ Dis the 0-cell; the complement to 1 in the boundary circle is the 1-cell, which wedenote by T ; and the interior of D is the 2-cell, which we denote by D. By takingproduct we obtain a cellular decomposition of Dm whose cells are parametrised bypairs of subsets J, I ⊂ [m] with J ∩ I = ∅: the set J parametrises the T -cells in

the product and I parametrises the D-cells. We denote the cell ofDm corresponding to a pair J, I by κ(J, I); it is a product of |J |cells of T -type and |I| cells of D-type (the positions in [m] \ I ∪Jare filled by 0-cells). Then ZK embeds as a cellular subcomplexin Dm; we have κ(J, I) ⊂ ZK whenever I ∈ K.

s1D

T

Let C∗(ZK) be the cellular cochains of ZK. It has a basis of cochains κ(J, I)∗

dual to the corresponding cells. We introduce the bigrading by setting

bidegκ(J, I)∗ = (−|J |, 2|I|+ 2|J |),so that bidegD = (0, 2), bidegT = (−1, 2) and bideg 1 = (0, 0). Since the cellulardifferential preserves the second grading, the complex C∗(ZK) splits into the sumof its components with fixed second degree:

(4.11) C∗(ZK) =

m⊕

q=0

C∗,2q(ZK).

The cohomology of the moment-angle complex therefore acquires an additionalgrading, and we define the bigraded Betti numbers of ZK by

(4.12) b−p,2q(ZK) = rankH−p,2q(ZK), for 1 6 p, q 6 m.

The ordinary Betti numbers of ZK therefore satisfy

(4.13) bk(ZK) =∑

−p+2q=k

b−p,2q(ZK).

The map of the moment-angle complexes ZK → ZL induced by a simplicialmap K → L (see Proposition 4.2.4) is clearly a cellular map. We therefore obtain

Proposition 4.4.1. The correspondence K 7→ ZK gives rise to a functor fromthe category of simplicial complexes and simplicial maps to the category of cellcomplexes with torus actions and equivariant maps. It also induces a natural trans-formation between the functor of simplicial cochains of K and the functor of cellularcochains of ZK.

The map ZK → ZL induced by a simplicial map K → L preserves the cellularbigrading, so that the bigraded cohomology groups are also functorial.


4.5. Cohomology ring

The main result of this section, Theorem 4.5.4, establishes an isomorphismbetween the integral cohomology ring of the moment-angle complex ZK and theTor-algebra of the simplicial complex K. This result was first proved in [53]for field coefficients using the Eilenberg–Moore spectral sequence of the fibrationZK → (CP∞)K → (CP∞)m. The proof given here is taken from [24] and [57]; itestablishes the isomorphism over the integers and works with the cellular cochains.

One of the key steps in the proof is the construction of a cellular approxima-tion of the diagonal map ∆ : ZK → ZK × ZK which is functorial with respect tomaps of moment-angle complexes induced by simplicial maps. The resulting cel-lular cochain algebra is isomorphic to the algebra R∗(K) from Construction 3.2.5(obtained by factorising the Koszul algebra of the face ring Z[K] by an acyclic ideal);its cohomology is isomorphic to the Tor-algebra of K.

Another proof of Theorem 4.5.4 was given by Franz [117].

Algebraic model for cellular cochains. We recall the algebra R∗(K) fromConstruction 3.2.5:

R∗(K) = Λ[u1, . . . , um]⊗ Z[K]/(v2i = uivi = 0, 1 6 i 6 m),

with the bigrading and differential given by

bideg ui = (−1, 2), bideg vi = (0, 2), dui = vi, dvi = 0.

The algebra R∗(K) has finite rank as an abelian group, with a basis of monomialsuJvI where J ⊂ [m], I ∈ K and J ∩ I = ∅.

Comparing the differential graded module structures in R∗(K) and C∗(ZK) weobserve that they coincide, as described in the following statement:

Lemma 4.5.1. The map

g : R∗(K)→ C∗(ZK), uJvI 7→ κ(J, I)∗,

is an isomorphism of cochain complexes. Hence, there is an additive isomorphism

H [R∗(K)] ∼= H∗(ZK).

Proof. Since g arises from a bijective correspondence between bases of R∗(K)and C∗(ZK), it is an isomorphism of bigraded modules (or groups). It also clearlycommutes with the differentials:

δg(ui) = δ(T ∗i ) = D∗

i = g(vi) = g(dui), δg(vi) = δ(D∗i ) = 0 = g(dvi),

where Ti denotes the cell κ(i,∅), and Di = κ(∅, i).

Having identified the algebra R∗(K) with the cellular cochains of the moment-angle complex, we can give a topological interpretation to the quasi-isomorphism ofLemma 3.2.6. To do this we shall identify the Koszul algebra Λ[u1, . . . , um]⊗Z[K]with the cellular cochains of a space homotopy equivalent to ZK.

The infinite-dimensional sphere S∞ is the direct limit (union) of standardlyembedded odd-dimensional spheres. Each odd sphere S2k+1 can be obtained fromS2k−1 by attaching two cells of dimensions 2k and 2k + 1:

S2k+1 ∼= (S2k−1 ∪f D2k) ∪g D2k+1.

Here the map f : ∂D2k → S2k−1 is the identity (and has degree 1), and the mapg : ∂D2k+1 = S2k → D2k is the projection of the standard sphere onto its equatorial

4.5. COHOMOLOGY RING 151

plane (and has degree 0). This implies that S∞ is contractible and has a celldecomposition with one cell in each dimension; the boundary of an even cell is theclosure of an odd cell, and the boundary of an odd cell is zero. The 2-dimensionalskeleton of this cell decomposition is the disc D2 decomposed into three cells asdescribed in Section 4.4. The cellular cochain complex of S∞ can be identified withthe Koszul algebra

Λ[u]⊗ Z[v], deg u = 1, deg v = 2, du = v, dv = 0.

The functoriality of the polyhedral product (Proposition 4.2.3 (a)) implies thatthere is the following deformation retraction onto a cellular subcomplex:

ZK = (D2, S1)K → (S∞, S1)K −→ (D2, S1)K.

Furthermore, the cellular cochains of the polyhedral product (S∞, S1)K are iden-tified with the Koszul algebra Λ[u1, . . . , um] ⊗ Z[K], in the same way as C∗(ZK)is identified with R∗(K). Since ZK ⊂ (S∞, S1)K is a deformation retract, thecorresponding cellular cochain map

Λ[u1, . . . , um]⊗ Z[K] = C∗((S∞, S1)K

)−→ C∗(ZK) = R∗(K)

induces an isomorphism in cohomology. The above map is a homomorphism of alge-bras, so it is a quasi-isomorphism. In fact, the cochain homotopy map constructedin the proof of Lemma 3.2.6 is nothing but the cellular cochain map induced by thehomotopy above.

Cellular diagonal approximation. Here we establish the cohomology ringisomorphism in Lemma 4.5.1. The difficulty of working with cellular cochains isthat they do not admit a functorial associative multiplication. The diagonal mapused in the definition of the cohomology product is not cellular, and a cellularapproximation cannot be made functorial with respect to arbitrary cellular maps.

Here we construct a canonical cellular diagonal approximation ∆ : ZK → ZK ×ZK

which is functorial with respect to maps of ZK induced by simplicial maps of K,and show that the resulting product in the cellular cochains of ZK coincides withthe product in R∗(K).

The product in the cohomology of a cell complex X is defined as follows(see [248], [151]). Consider the composite map of cellular cochain complexes

(4.14) C∗(X)⊗ C∗(X)×−−−−→ C∗(X ×X)

∆∗

−−−−→ C∗(X).

Here the map × sends a cellular cochain c1 ⊗ c2 ∈ Cq1(X)⊗ Cq2(X) to the cochainc1×c2 ∈ Cq1+q2(X×X), whose value on a cell e1×e2 ∈ X×X is (−1)q1q2c1(e1)c2(e2).The map ∆∗ is induced by a cellular map ∆ (a cellular diagonal approximation)homotopic to the diagonal ∆ : X → X ×X . In cohomology, map (4.14) induces amultiplication H∗(X) ⊗ H∗(X) → H∗(X) which does not depend on a choice ofcellular approximation and is functorial. However, map (4.14) itself is not functorialbecause the choice of a cellular approximation is not canonical.

Nevertheless, in the case X = ZK we can use the following construction.

Construction 4.5.2 (cellular approximation for ∆ : ZK → ZK × ZK). Con-

sider the map ∆ : D→ D×D given in the polar coordinates z = ρeiϕ ∈ D, 0 6 ρ 6 1,0 6 ϕ < 2π, by the formula

(4.15) ρeiϕ 7→

(1− ρ+ ρe2iϕ, 1) for 0 6 ϕ 6 π,(1, 1− ρ+ ρe2iϕ) for π 6 ϕ < 2π.


It is easy to see that this is a cellular map homotopic to the diagonal∆ : D→ D×D,and its restriction to the boundary circle S is a diagonal approximation for S, asdescribed by the following diagram:

(4.16)

S −−−−→ D

∆

yy∆

S× S −−−−→ D× D

(explicit formulae for the homotopies involved can be found in Exercise 4.5.11).

Taking the m-fold product we obtain a cellular approximation ∆ : Dm → Dm×Dm.

Applying Proposition 4.2.3 (a) to the map of pairs ∆ : (D, S)→ (D×D, S× S) and

observing that (D × D, S × S)K ∼= ZK × ZK, we obtain that ∆ : Dm → Dm × Dm

restricts to a cellular approximation of the diagonal map of ZK, as described in thefollowing diagram:

ZK −−−−→ Dm

∆

yy∆

ZK ×ZK −−−−→ Dm × Dm.Finally, applying Proposition 4.2.3 (c) to diagram (4.16) we obtain that the ap-

proximation ∆ is functorial with respect to the maps of moment-angle-complexesZK → ZL induced by simplicial maps K → L.

Lemma 4.5.3. The cellular cochain algebra C∗(ZK) with the product defined via

the diagonal approximation ∆ : ZK → ZK × ZK and map (4.14) is isomorphic tothe algebra R∗(K). We therefore have an isomorphism of cohomology rings

H [R∗(K)] ∼= H∗(ZK).

Proof. We first consider the case K = ∆0, i.e. ZK = D. The cellular cochaincomplex has basis of cochains 1 ∈ C0(D), T ∗ ∈ C1(D) and D∗ ∈ C2(D) dual to thecells introduced in Section 4.4. The multiplication defined by (4.14) in C∗(D) istrivial, so we have a ring isomorphism

R∗(∆0) = Λ[u]⊗ Z[v]/(v2 = uv = 0) −→ C∗(D).

Taking an m-fold tensor product we obtain a ring isomorphism for K = ∆m−1:

f : R∗(∆m−1) = Λ[u1, . . . , um]⊗ Z[v1, . . . , vm]/(v2i = uivi = 0) −→ C∗(Dm).

Now for arbitrary K we have an inclusion ZK ⊂ Dm = Z∆m−1 of a cellu-lar subcomplex and the corresponding ring homomorphism q : C∗(Dm) → C∗(ZK).Consider the commutative diagram

R∗(∆m−1)f−−−−→ C∗(Dm)

p

yyq

R∗(K) g−−−−→ C∗(ZK).

Here the maps p, f and q are ring homomorphisms, and g is an isomorphism ofgroups by Lemma 4.5.1. We claim that g is also a ring isomorphism. Indeed, takeα, β ∈ R∗(K). Since p is onto, we have α = p(α′) and β = p(β′). Then

g(αβ) = gp(α′β′) = qf(α′β′) = qf(α′)qf(β′) = gp(α′)gp(β′) = g(α)g(β),

as claimed. Thus, g is a ring isomorphism.

4.5. COHOMOLOGY RING 153

Main result. By combining the results of Lemmata A.2.10, 3.2.6 and 4.5.3we obtain the main result of this section:

Theorem 4.5.4. There are isomorphisms, functorial in K, of bigraded algebras

H∗,∗(ZK) ∼= TorZ[v1,...,vm]

(Z[K],Z

)

∼= H(Λ[u1, . . . , um]⊗ Z[K], d

),

where the bigrading and the differential on the right hand side are defined by

bideg ui = (−1, 2), bideg vi = (0, 2), dui = vi, dvi = 0.

The algebraic Betti numbers (3.2) of the face ring Z[K] therefore acquire atopological interpretation as the bigraded Betti numbers (4.12) of the moment-angle complex ZK.

Now we combine results of Propositions 3.1.5, 3.2.2 and 4.4.1, Corollary 4.3.3and Theorem 4.5.4 in the following statement describing the functorial propertiesof the correspondence K 7→ ZK.

Proposition 4.5.5. Consider the following functors:

(a) Z, the covariant functor K 7→ ZK from the category of finite simplicialcomplexes and simplicial maps to the category of spaces with torus actionsand equivariant maps (the moment-angle complex functor);

(b) k[·], the contravariant functor K 7→ k[K] from simplicial complexes tograded k-algebras (the face ring functor);

(c) Tor-alg, the contravariant functor

K 7→ Tork[v1,...,vm]

(k[K],k

)

from simplicial complexes to bigraded k-algebras (the Tor-algebra functor;it is the composition of k[·] and Tork[v1,...,vm](· ,k));

(d) H∗T , the contravariant functor X 7→ H∗

T (X ;k) from spaces with torusactions to k-algebras (the equivariant cohomology functor);

(e) H∗, the contravariant functor X 7→ H∗(X ;k) from spaces to k-algebras(the ordinary cohomology functor).

Then we have the following identities:

H∗T Z = k[·], H∗ Z = Tor-alg.

The second identity implies that for any simplicial map ϕ : K → L the correspondingcohomology map

ϕ∗Z : H∗(ZL)→ H∗(ZK)

coincides with the induced homomorphism of Tor-algebras ϕ∗Tor from Proposi-

tion 3.2.2. In particular, the map ϕ gives rise to a map

H−q,2p(ZL)→ H−q,2p(ZK)

of bigraded cohomology.

In the case of Cohen–Macaulay complexes K (see Section 3.3) we have thefollowing version of Theorem 4.5.4.

Proposition 4.5.6. Let K be an (n−1)-dimensional Cohen–Macaulay complex,and let t be an hsop in k[K]. Then we have the following isomorphism of algebras:

H∗(ZK;k) ∼= Tork[v1,...,vm]/t

(k[K]/t,k

).


Proof. This follows from Theorem 4.5.4 and Lemma A.3.5.

Note that the algebra k[K]/t is finite-dimensional as a k-vector space, unlikek[K]. In some circumstances this observation allows us to calculate the cohomologyof ZK more effectively.

Description of the product in terms of full subcomplexes. TheHochster formula (Theorem 3.2.4) for the components of the Tor-algebra can beused to obtain an alternative description of the product structure in H∗(ZK).

We recall from Section 3.2 that the bigraded structure in the Tor-algebra isrefined to a multigrading, and the multigraded components of Tor can be calculatedin terms of the full subcomplexes of K:

Tor−i,2JZ[v1,...,vm]

(Z[K],Z

) ∼= H |J|−i−1(KJ ),where J ⊂ [m], see Theorem 3.2.9. Furthermore, the product in the Tor-algebra

defines a product in the direct sum⊕

p>0, J⊂[m] Hp−1(KJ ) given by (3.11).

The bigraded structure in the cellular cochain complex of ZK defined in Sec-tion 4.4 can be also refined to a multigrading (a Z⊕ Zm-grading):

C∗(ZK) =⊕

J⊂[m]

C∗, 2J (ZK),

where C∗, 2J(ZK) is the subcomplex spanned by the cochains κ(J\I, I)∗ with I ⊂ Jand I ∈ K. The bigraded cohomology groups are decomposed as follows:

H−i, 2j(ZK) =⊕

J⊂[m] : |J|=j

H−i, 2J(ZK),

where H−i, 2J(ZK) = H−i[C∗, 2J(ZK)].

Theorem 4.5.7 (Baskakov [22]). There are isomorphisms

Hp−1(KJ )∼=−→ Hp−|J|,2J(ZK),

which are functorial with respect to simplicial maps and induce a ring isomorphism

h :∑

J⊂[m]

H∗(KJ )∼=−→ H∗(ZK).

Proof. The statement about the additive isomorphisms follows from Theo-rems 3.2.9 and 4.5.4. In explicit terms, the cohomology isomorphisms are inducedby the cochain isomorphisms given by

Cp−1(KJ ) −→ Cp−|J|,2J(ZK),

αL 7−→ ε(L, J)κ(J \ L,L)∗

similar to (3.9), where αL ∈ Cp−1(KJ ) is the cochain dual to a simplex L ∈ KJ .The ring isomorphism follows from Proposition 3.2.10 and Theorem 4.5.4.

We summarise the results above in the following description of the cohomologygroups and the product structure of H∗(ZK) in terms of full subcomplexes of K:

Theorem 4.5.8. There are isomorphisms of groups

H−i,2j(ZK) ∼=⊕

J⊂[m] : |J|=j

Hj−i−1(KJ ), Hℓ(ZK) ∼=⊕

J⊂[m]

Hℓ−|J|−1(KJ ).

4.6. BIGRADED BETTI NUMBERS 155

These isomorphisms sum up into a ring isomorphism

H∗(ZK) ∼=⊕

J⊂[m]

H∗(KJ ),

where the ring structure on the right hand side is given by the canonical maps

Hk−|I|−1(KI)⊗Hℓ−|J|−1(KJ)→ Hk+ℓ−|I|−|J|−1(KI∪J )which are induced by simplicial maps KI∪J → KI ∗ KJ for I ∩ J = ∅ and zerootherwise.

It follows that the cohomology of ZK may have arbitrary torsion:

Corollary 4.5.9. Any finite abelian group can appear as a summand in acohomology group of H∗(ZK) for some K.

Proof. It follows from the Theorem 4.5.8 that H∗(K) is a direct summandin H∗(ZK) (with appropriate shifts in dimension). Therefore, we can take K whosesimplicial cohomology contains the appropriate torsion.

Exercises.

4.5.10. Let S be the standard unit circle decomposed into two cells, where the0-cell is the unit. The map

∆ : S→ S× S, eiϕ 7→

(e2iϕ, 1) for 0 6 ϕ 6 π,(1, e2iϕ) for π 6 ϕ < 2π

is a cellular diagonal approximation. It is obtained by restricting map (4.15) tothe boundary circle (ρ = 1). A homotopy Ft between the diagonal ∆ : S → S × S

(t = 0) and its cellular approximation ∆ (t = 1) is given by

Ft : S→ S× S, eiϕ 7→ (

ei(1+t)ϕ, ei(1−t)ϕ)

for 0 6 ϕ 6 π,(ei(1−t)ϕ+2πit, ei(1+t)ϕ−2πit

)for π 6 ϕ < 2π.

4.5.11. Show that the formula

ρeiϕ 7→ (

(1 − ρ)t+ ρei(1+t)ϕ, (1− ρ)t+ ρei(1−t)ϕ)

for 0 6 ϕ 6 π,((1 − ρ)t+ ρei(1−t)ϕ+2πit, (1− ρ)t+ ρei(1+t)ϕ−2πit

)for π 6 ϕ < 2π.

defines a homotopy Gt : D → D × D between the diagonal ∆ : D → D × D (t = 0)

and its approximation ∆ (4.16) (t = 1). For ρ = 1 the homotopy Gt restricts tothe homotopy Ft of the previous exercise.

4.6. Bigraded Betti numbers

Here we describe the main properties of the bigraded Betti numbers (4.12) ofmoment-angle complexes and give some examples of explicit calculations.

Lemma 4.6.1. Let K be a simplicial complex of dimension n− 1 with f0 = mvertices and f1 edges, so that dimZK = m+ n. We have

(a) b0,0(ZK) = b0(ZK) = 1 and b0, 2q(ZK) = 0 for q 6= 0;(b) b−p, 2q = 0 for q > m or p > q;(c) b1(ZK) = b2(ZK) = 0;(d) b3(ZK) = b−1,4(ZK) =

(m2

)− f1;

(e) b−p, 2q(ZK) = 0 for p > q > 0 or q − p > n;


(f) bm+n(ZK) = b−(m−n), 2m(ZK) = rank Hn−1(K).Proof. We consider the algebra R∗(K) whose cohomology is H∗(ZK). Recall

that R∗(K) has additive basis of monomials uJvI with I ∈ K and I ∩ J = ∅. Sincebideg vi = (0, 2), bideg uj = (−1, 2), the bigraded component R−p,2q(K) has basisof monomials uJvI with |I| = q − p and |J | = p. In particular, R−p,2q(K) = 0 forq > m or p > q, which implies (b). To prove (a) we observe that R0,0(K) = k andeach vI ∈ R0,2q(K) with q > 0 is a coboundary, hence, H0,2q(ZK) = 0 for q > 0.

Now we prove (e). Let uJvI ∈ R−p,2q(K); then |I| = q − p and I ∈ K. Sincea simplex of K has at most n vertices, R−p,2q(K) = 0 for q − p > n. We haveb−p,2q(ZK) = 0 for p > q by (b) so we need only to check that b−q,2q(ZK) = 0 forq > 0. The group R−q,2q(K) has basis of monomials uJ with |J | = q. Since d(ui) =vi, there are no nonzero cocycles in R−q,2q(K) for q > 0, hence, H−q,2q(ZK) = 0.

Statement (c) follows from (e) and (4.13).We also have H3(ZK) = H−1,4(ZK), by (e). There is a basis in R−1,4(K)

consisting of monomials ujvi with i 6= j. We have d(ujvi) = vivj and d(uiuj) =ujvi − uivj . Hence, ujvi is a cocycle if and only if i, j /∈ K; in this case the twococycles ujvi and uivj represent the same cohomology class. This proves (d).

It remains to prove (f). The total degree of a monomial uJvI ∈ R∗(K) is2|I| + |J |, and there are constraints |I| + |J | 6 m and |I| 6 n. Therefore, themaximum of the total degree is achieved for |I| = n and |J | = m− n. This provesthe first identity of (f), and the second follows from Theorem 4.5.8.

Lemma 4.6.1 shows that nonzero bigraded Betti numbers br,2q(ZK) with r 6= 0appear only in the strip bounded by the lines r = −1, q = m, r + q = 1 andr + q = n in the second quadrant, see Fig. 4.1 (a).

0

2

4

...

2m

0−1. . .−(m − n)−m

1

∗∗

∗∗

∗∗

∗

∗∗

∗∗

∗∗

∗∗

∗∗

∗

∗∗

∗∗

(a) arbitrary K

0

2

4

...

2m

0−1. . .−(m − n)−m

1

∗∗

∗∗

∗∗

∗∗

∗

1

(b) |K| ∼= Sn−1

Figure 4.1. Possible locations on nonzero b−p,2q(ZK) (marked by ∗).

The next result allows us to express the numbers of faces of K (i.e. its f - andh-vectors) via the bigraded Betti numbers. We consider the Euler characteristicsof the complexes C∗,2q(ZK) (see (4.11)),

(4.17) χq(ZK) =

m∑

p=0

(−1)p rank C−p,2q(ZK) =

m∑

p=0

(−1)pb−p,2q(ZK)


and define the generating series

χ(ZK; t) =

m∑

q=0

χq(ZK)t2q.

Theorem 4.6.2. The following identity holds for an (n− 1)-dimensional sim-plicial complex K with m vertices:

χ(ZK; t) = (1 − t2)m−n(h0 + h1t2 + · · ·+ hnt

2n).

Here (h0, h1, . . . , hn) is the h-vector of K.

Proof. The bigraded component C−p,2q(ZK) has basis of cellular cochainsκ(J, I)∗ with I ∈ K, |I| = q − p and |J | = p. Therefore, rank C−p,2q(ZK) =fq−p−1

(m−q+p

p

), where (f0, f1, . . . , fn−1) is the f -vector of K and f−1 = 1. By

substituting this into (4.17) we obtain

χq(ZK) =

m∑

j=0

(−1)q−jfj−1

(m−jq−j

).

Then

(4.18) χ(ZK; t) =m∑

q=0

m∑

j=0

t2jt2(q−j)(−1)q−jfj−1

(m−jq−j

)

=m∑

j=0

fj−1t2j(1 − t2)m−j = (1− t2)m

n∑

j=0

fj−1(t−2 − 1)−j.

Set h(t) = h0 + h1t+ · · ·+ hntn. Then it follows from (2.3) that

tnh(t−1) = (t− 1)nn∑

j=0

fj−1(t− 1)−j .

By substituting t−2 for t in the identity above we finally rewrite (4.18) as

χ(ZK; t)

(1− t2)m =t−2nh(t2)

(t−2 − 1)n=

h(t2)

(1− t2)n ,

which is equivalent to the required identity.

Corollary 4.6.3. If K 6= ∆m−1, then the Euler characteristic of ZK is zero.

Proof. We have

χ(ZK) =m∑

p,q=0

(−1)−p+2qb−p,2q(ZK) =m∑

q=0

χq(ZK) = χ(ZK; 1) = 0

by Theorem 4.6.2 (note that K 6= ∆m−1 implies that m > n).

We proceed by describing the properties of bigraded Betti numbers for partic-ular classes of simplicial complexes.

Definition 4.6.4. A finite simplicial complex K is called a d-dimensional pseu-domanifold if the following three conditions are satisfied:

(a) all maximal simplices of K have dimension d (i.e. K is pure d-dimensional);(b) each (d− 1)-simplex of K is the face of exactly two d-simplices of K.


(c) if I and I ′ are d-simplices of K, then there is a sequence I = I1, I2, . . . , Ik =I ′ of d-simplices of K such that Ij and Ij+1 have a common (d − 1)-facefor 1 6 i 6 k − 1.

If K is a d-dimensional pseudomanifold, then either Hd(K) ∼= Z or 0 (an exer-cise). In the former case the pseudomanifold K is called orientable.

Lemma 4.6.5. Let K be an orientable pseudomanifold of dimension n− 1 withm vertices. Then

Hm+n(ZK) = Hn−1(K) ∼= Z.

Under the isomorphism H∗(ZK) ∼= H [R∗(K)], the group above is generated by theclass of any monomial uJvI ∈ R∗(K) of bidegree (−(m − n), 2m) such that I ∈ Kand J = [m] \ I.

Proof. The isomorphism of groups follows from Theorem 4.5.8 and the factthat K is orientable. We have Hm+n(ZK) = H−(m−n),2m(ZK). The groupR−(m−n),2m(K) has basis of monomials uJvI with I ∈ K, |I| = n and J = [m] \ I.Each of these monomials is a cocycle. Let I, I ′ be two (n − 1)-simplices of K hav-ing a common (n − 2)-face. Consider the corresponding cocycles uJvI and uJ′vI′

(where J = [m] \ I, J ′ = [m] \ I ′):uJvI = uj1uj2 · · ·ujm−n

vi1 · · · vin−1vin ,

uJ′vI′ = uinuj2 · · ·ujm−nvi1 · · · vin−1vj1 .

Since K is a pseudomanifold, the (n − 2)-face i1, . . . , in−1 is contained in ex-actly two (n − 1)-faces, namely I = i1, . . . , in−1, in and I ′ = i1, . . . , in−1, j1.Therefore we have the following identity in R∗(K)

d(uinuj1uj2 · · ·ujm−nvi1 · · · vin−1) =

= uj1uj2 · · ·ujm−nvi1 · · · vin−1vin − uinuj2 · · ·ujm−n

vi1 · · · vin−1vj1

Hence, [uJvI ] = [uJ′vI′ ] (as cohomology classes). Property (c) from the definitionof a pseudomanifold implies that all monomials uJvI ∈ R−(m−n),2m(K) representthe same cohomology class up to sign. The isomorphism (3.9) takes uJvI to ±αI ∈Cn−1(K), which represents a generator of Hn−1(K) ∼= Z (see Exercise 4.6.17).

Remark. If K is a non-orientable pseudomanifold, then the same argumentshows that any monomial uJvI ∈ R∗(K) as above represents the generator ofHm+n(ZK) = Hn−1(K) ∼= Z2.

Proposition 4.6.6. Let K be a triangulated sphere of dimension n− 1. ThenPoincare duality for the moment-angle manifold ZK respects the bigrading in coho-mology. In particular,

b−p, 2q(ZK) = b−(m−n)+p, 2(m−q)(ZK) for 0 6 p 6 m− n, 0 6 q 6 m.

Proof. The Poincare duality maps (see Definition 3.4.3) are defined via thecohomology multiplication in H∗(ZK), which respects the bigrading. We havedimZK = m+ n, and

Hm+n(ZK) = Tor−(m−n), 2mZ[v1,...,vm]

(Z[K],Z

) ∼= Z,

by Lemma 4.6.5. This implies the required identity for the Betti numbers.


Corollary 4.6.7. Let K be a triangulated (n − 1)-sphere and ZK the corre-sponding moment-angle manifold, dimZK = m+ n. Then

(a) b−p,2q(ZK) = 0 for p > m− n, with the only exception b−(m−n),2m = 1;(b) b−p,2q(ZK) = 0 for q − p > n, with the only exception b−(m−n),2m = 1.

It follows that if |K| ∼= Sn−1, then nonzero bigraded Betti numbers br,2q(ZK),except b0,0(ZK) and b−(m−n),2m(ZK), appear only in the strip bounded by the linesr = −(m− n− 1), r = −1, r + q = 1 and r + q = n− 1 in the second quadrant,see Fig. 4.1 (b).

A space X is called a Poincare duality space (over k) if H∗(X ;k) is a Poincarealgebra (see Definition 3.4.3). We have the following characterisation of moment-angle complexes with Poincare duality, extending the result of Corollary 4.6.6.

Theorem 4.6.8. ZK is a Poincare duality space over a field k if and only K isa Gorenstein complex over k.

Proof. Assume that K is a Gorenstein complex. Consider the algebra T

defined in Theorem 3.4.4, i.e. T =⊕d

i=0 Ti, where T i = Tor−i

k[m](k[K],k) and

d = maxj : Tor−jk[m](k[K],k) 6= 0. Since T is Poincare algebra, k ∼= T 0 ∼=

Homk(Td, T d), which implies that T d ∼= k. Since T has a bigrading, we obtain

T d = T d,2q for some q > 0. Since the multiplication in T respects the bigrading,

the isomorphisms T i∼=→ Homk(T

d−i, T d) from the definition of a Poincare algebrasplit into isomorphisms

T i, 2j∼=−→ Homk(T

d−i, 2(q−j), T d,2q).

Let Hk = Hk(ZK;k) and H =⊕r

k=0Hk; then Hk =

⊕i+2j=k T

i, 2j and r = d+2q.Therefore, we have the isomorphisms

Hk =⊕

i+2j=k

T i,2j∼=−→

⊕

i+2j=k

Homk(Td−i,2(q−j), T d,2q) = Homk(H

r−k, Hr),

which imply that H is a Poincare algebra.Now assume that H =

⊕rk=0H

k is a Poincare algebra. Then

k ∼= Hr = Tor−d, 2qk[v1,...,vm]

(k[K],k

)= T d, 2q

for some d, q > 0. Since the multiplication in the cohomology of ZK respects the

bigrading, the isomorphisms Hk∼=→ Homk(H

r−k, Hr) split into isomorphisms

H−i, 2j = T i, 2j∼=−→ Homk(T

d−i, 2(q−j), T d, 2q),

which in their turn define the isomorphisms

T i =⊕

j

T i, 2j∼=−→⊕

j

Homk(Td−i, 2(q−j), T d) = Homk(T

d−i, T d).

Thus, T is a Poincare algebra.

Remark. We do not assume that r = maxk : Hk(ZK;k) 6= 0 is equal todimZK = m + n in Theorem 4.6.8. It follows from the proof above that ZK is aPoincare duality space with r = dimZK if and only if K is a Gorenstein* complex.

Here are some explicit examples of calculations ofH∗(ZK) using Theorem 4.5.4.


Example 4.6.9. Let K = ∂∆m−1. Then

Z[K] = Z[v1, . . . , vm]/(v1 · · · vm).

The cocycle u1v2v3 · · · vm ∈ Λ[u1, . . . , um]⊗Z[K] of bidegree (−1, 2m) represents agenerator of the top degree cohomology group of ZK

∼= S2m−1.

Example 4.6.10. Let K be the boundary of 5-gon. We have dimZK = 7.We enumerate the vertices of K clockwise. The face ring of K is given in Ex-ample 3.1.4.3. The group H3(ZK) has 5 generators corresponding to the diag-onals of the 5-gon; these generators are represented by the cocycles uivi+2 ∈Z[K] ⊗ Λ[u1, . . . , u5], 1 6 i 6 5 (the summation of indices is modulo 5). A directcalculation shows that H4(ZK) also has 5 generators, represented by the cocyclesujuj+1vj+3, 1 6 j 6 5. The Betti vector of ZK is therefore given by

(b0(ZK), b

1(ZK), . . . , b7(ZK)

)= (1, 0, 0, 5, 5, 0, 0, 1).

By Lemma 4.6.5, the product of cocycles uivi+2 and ujuj+1vj+3 represents a gen-erator of H7(ZK) if and only if all the indices i, i + 2, j, j + 1, j + 3 are differ-ent. Hence, for each cohomology class [uivi+2] ∈ H3(ZK) there is a unique class[ujuj+1vj+3] ∈ H4(ZK) such that the product [uivi+2] · [ujuj+1vj+3] is nonzero.These calculations are summarised by the cohomology ring isomorphism

H∗(ZK) ∼= H∗((S3 × S4)#5

),

in accordance with Exercise 4.2.10.

Example 4.6.11. Now we calculate the Betti numbers and the cohomologyproduct for ZK in the case when K is a boundary of an m-gon with m > 4.

It follows from Corollary 4.6.7 that the only nonzero bigraded Betti numbersof ZK are b−(p−1),2p for 2 6 p 6 m− 2 and b0,0(ZK) = b−(m−2),2m(ZK) = 1. Theordinary Betti numbers are therefore given by

b0(ZK) = bm+2(ZK) = 1, bk(ZK) = b−(k−2),2(k−1)(ZK) for 3 6 k 6 m− 1.

To calculate b−(k−2),2(k−1)(ZK) for 3 6 k 6 m− 1 we use the algebra R∗(K) ofConstruction 3.2.5. We have

(4.19) b−(k−2),2(k−1)(ZK) = rankH−(k−2),2(k−1)(R∗, d) =

rankker[d : R−(k−2),2(k−1) → R−(k−3),2(k−1)

]− rankdR−(k−1),2(k−1).

Since H−(k−1),2(k−1)(ZK) = 0 for k > 1, the differential d from R−(k−1),2(k−1) ismonomorphic, and

rank dR−(k−1),2(k−1) = rankR−(k−1),2(k−1).

Similarly, since H−(k−3),2(k−1)(ZK) = 0 for k 6 m, the differential fromR−(k−2),2(k−1) is epimorphic, and

rank ker[d : R−(k−2),2(k−1) → R−(k−3),2(k−1)

]

= rankR−(k−2),2(k−1) − rankR−(k−3),2(k−1).


Substituting the above two expressions into (4.19) and using formula (3.3) for thedimensions of R−p,2q, we calculate

bk(ZK) = b−(k−2),2(k−1)(ZK)

= rankR−(k−2),2(k−1) − rankR−(k−3),2(k−1) − rankR−(k−1),2(k−1)

= m(m−1k−2

)−m

(m−2k−3

)−(mk−1

)

= (k − 2)(m−2k−1

)+ (m− k)

(m−2

m−k+1

)for 3 6 k 6 m− 1.

Note that the cohomology of ZK does not have torsion in this example (thisfollows from Theorem 4.5.8).

The product of any three classes of positive degree in H∗(ZK) is zero (i.e.the cohomology product length of ZK is 2). Indeed, if αi ∈ H−(pi−1),2pi(ZK) fori = 1, 2, 3, then

α1α2α3 ∈ H−(p1+p2+p3−3),2(p1+p2+p3)(ZK),

which is zero by Lemma 4.6.1 (e) (note that n = 2 in this example). Hence allnontrivial products in H∗(ZK) arise from Poincare duality.

The above calculations of the Betti numbers together with the observationsabout the cohomology product can be summarised by saying that ZK cohomologi-cally looks like a connected sum of sphere products, namely,

(4.20) H∗(ZK) ∼= H∗(m−1

#k=3

(Sk × Sm+2−k

)#(k−2)(m−2k−1

)),

as rings.

According to a result of McGavran [216], the cohomology isomorphism of (4.20)is induced by a homeomorphism of manifolds. This is true even in a more generalsituation, when K is the boundary of a stacked polytope (see Definition 1.4.8):

Theorem 4.6.12 ([216], [36, Theorem 6.3]). Let K be the boundary of a stackedpolytope of dimension n with m > n+1 vertices. Then the corresponding moment-angle manifold is homeomorphic to a connected sum of sphere products,

ZK∼=

m−n+1

#k=3

(Sk × Sm+n−k

)#(k−2)(m−nk−1

).

For n = 2 we obtain a homeomorphism of manifolds underlying the isomor-phism (4.20) (note that any 2-polytope is stacked).

The bigraded Betti numbers for stacked polytopes can also be calculated:

Theorem 4.6.13 ([303], [72], [195]). Let K be as in Theorem 4.6.12 and n > 3.Then the nonzero bigraded Betti numbers of ZK are given by

b0,0(ZK) = b−(m−n),2m(ZK) = 1,

b−i,2(i+1)(ZK) = i(m−ni+1

)for 1 6 i 6 m− n− 1,

b−i,2(i+n−1)(ZK) = (m− n− i)(

m−nm−n+1−i

)for 1 6 i 6 m− n− 1.

Exercises.

4.6.14. A triangulated manifold K is a pseudomanifold.

4.6.15. If K is a d-dimensional pseudomanifold, then either Hd(K) ∼= Zor Hd(K) ∼= 0. What happens for homology and cohomology with coefficientsin a field k?


4.6.16. If K is an orientable d-dimensional pseudomanifold (i.e. Hd(K) ∼= Z)then Hd(K) ∼= Z, and if K is non-orientable then Hd(K) ∼= Z2. What happens forcohomology with coefficients in a field k?

4.6.17. Let K be an orientable d-dimensional pseudomanifold. The homologygroupHd(K) ∼= Z is generated by the class of simplicial chain 〈K〉 =∑I∈K, dim I=d I

where the d-simplices I ∈ K are oriented properly (the fundamental homology classofK). The cohomology groupHd(K) ∼= Z is generated by the class of any cochain αItaking value 1 on an oriented d-simplex I ∈ K and vanishing on all other simplices.

4.6.18. Calculate H∗(ZK) where K is the boundary of a pentagon using Theo-rem 4.5.8 and the description of the cohomology product given in Proposition 3.2.10(or Exercise 3.2.14).

4.7. Coordinate subspace arrangements

Here we establish a homotopy equivalence between the moment-angle complexZK and the complement of the arrangement of coordinate subspaces in Cm corre-sponding to a simplicial complex K. As a corollary we obtain an explicit descriptionof the cohomology ring of a coordinate subspace arrangement complement. In somecases, knowing the cohomology ring allows us to identify the homotopy type of ar-rangement complements.

Coordinate subspace arrangements already appeared in Section 3.1 as affinealgebraic varieties corresponding to face rings (see Proposition 3.1.12). Here weconsider these arrangements from the general point of view.

A coordinate subspace in Cm can be given as

(4.21) LI = (z1, . . . , zm) ∈ Cm : zi1 = · · · = zik = 0,where I = i1, . . . , ik is a subset of [m].

Construction 4.7.1. We assign to a simplicial complex K the arrangementof complex coordinate subspaces (or coordinate subspace arrangement) given by

A(K) = LI : I /∈ K.We denote by U(K) the complement to A(K) in Cm, that is,

(4.22) U(K) = Cm \⋃

I /∈K

LI .

Observe that if K′ ⊂ K is a subcomplex, then U(K′) ⊂ U(K).Proposition 4.7.2. The assignment K 7→ U(K) defines a bijective inclusion-

preserving correspondence between simplicial complexes on the set [m] and comple-ments of coordinate subspace arrangements in Cm.

Proof. We need to reconstruct a simplicial complex from the complementand check that it indeed defines the inverse correspondence. Let A be a coordinatesubspace arrangement in Cm, and let U be its complement. Set

K(U) = I ⊂ [m] : LI ∩ U 6= ∅.It is easy to see that K(U) is a simplicial complex satisfying U(K(U)) = U andK(U(K)) = K.

4.7. COORDINATE SUBSPACE ARRANGEMENTS 163

If i is a ghost vertex of K, then the coordinate subspace arrangement A(K)contains the hyperplane zi = 0. The arrangement A(K) does not contain hyper-planes if and only the vertex set of K is the whole [m].

The complement U(K) is an example of a polyhedral product space (see Con-struction 4.2.1), as is shown by the next proposition.

Proposition 4.7.3. U(K) = (C,C×)K.

Proof. Given a point z = (z1, . . . , zm) ∈ Cm, consider its zero set ω(z ) =i ∈ [m] : zi = 0 ⊂ [m]. We have

U(K) = Cm \⋃

I /∈K

LI = Cm \⋃

I /∈K

z : ω(z ) ⊃ I = Cm \⋃

I /∈K

z : ω(z ) = I

=⋃

I∈K

z : ω(z ) = I =⋃

I∈K

z : ω(z ) ⊂ I =⋃

I∈K

(C,C×)I = (C,C×)K.

Example 4.7.4.1. If K = ∆m−1, then U(K) = Cm.2. If K = ∂∆m−1, then U(K) = Cm \ 0.3. Let K be the discrete complex with m vertices. Then

U(K) = Cm \⋃

16i<j6m

zi = zj = 0

is the complement to all coordinate subspaces of codimension two.4. More generally, if K is the i-dimensional skeleton of ∆m−1, then U(K) is the

complement to all coordinate subspaces of codimension i+ 2.

Since each coordinate subspace is invariant under the standard action of Tm

on Cm, the complement U(K) is also a Tm-invariant subset in Cm.A deformation retraction of a space X onto a subspace A is a homotopy

Ft : X → X , t ∈ I, such that F0 = id (the identity map), F1(X) = A and Ft|A = idfor all t. The term ‘deformation retraction’ is often used only for the last mapf = F1 : X → A; this map is a homotopy equivalence.

Theorem 4.7.5. The moment-angle complex ZK is a Tm-invariant subspaceof U(K), and there is a Tm-equivariant deformation retraction

ZK → U(K) ≃−→ ZK.

Proof. We have ZK = (D, S)K ⊂ (C,C×)K = U(K) by the functoriality of thepolyhedral product, so the moment-angle complex ZK is indeed contained in thecomplement U(K) as a Tm-invariant subset.

The deformation retraction U(K)→ ZK will be constructed by induction. Weremove simplices from ∆m−1 until we obtain K, in such a way that we get a sim-plicial complex at each intermediate step.

The base of induction is clear: if K = ∆m−1, then U(K) = Cm, ZK = Dm, andthe retraction Cm → Dm is evident.

The orbit space ZK/Tm is the cubical complex cc(K) = (I, 1)K (see Construc-tion 2.9.11). The orbit space U(K)/Tm can be identified with

U(K)> = U(K) ∩ Rm> = (R>,R>)K

where Rm> is viewed as a subset in Cm.


We shall first construct a deformation retraction r : U(K)> → cc(K) of orbitspaces, and then cover it by a deformation retraction r : U(K)→ ZK.

Now assume that K is obtained from a simplicial complex K′ by removing onemaximal simplex J = j1, . . . , jk, i.e. K ∪ J = K′. Then the cubical complexcc(K′) is obtained from cc(K) by adding a single k-dimensional face CJ = (I, 1)J .We also have U(K) = U(K′) \ LJ , so that

U(K)> = U(K′)> \ y : yj1 = · · · = yjk = 0.

We may assume by induction that there is a deformation retraction r′ : U(K′)> →cc(K′) such that ω(r′(y)) = ω(y), where ω(y) is the set of zero coordinates of y .In particular, r′ restricts to a deformation retraction

r′ : U(K′)> \ y : yj1 = · · · = yjk = 0 −→ cc(K′)\ yJwhere yJ is the point with coordinates yj1 = · · · = yjk = 0 and yj = 1 for j /∈ J .

Since J /∈ K, we have yJ /∈ cc(K). On the other hand, yJ belongs to the extraface CJ = (I, 1)J of cc(K′). We therefore may apply the deformation retraction rJshown in Fig. 4.2 on the face CJ , with centre at yJ . In coordinates, a homotopy Ft

yJ

Figure 4.2. Retraction rJ : cc(K′)\ yJ → cc(K).

between the identity map cc(K′)\ yJ → cc(K′)\ yJ (for t = 0) and the retractionrJ : cc(K′)\ yJ → cc(K) (for t = 1) is given by

Ft : cc(K′)\ yJ −→ cc(K′)\ yJ ,(y1, . . . , ym, t) 7−→ (y1 + tα1y1, . . . , ym + tαmym)

where

αi =

1−maxj∈J yjmaxj∈J yj

, if i ∈ J ,

0, if i /∈ J ,for 1 6 i 6 m.

We observe that ω(Ft(y)) = ω(y) for any t and y ∈ cc(K′). Now, the composition

(4.23) r : U(K)> = U(K′)>\ y : yj1 = · · · = yjk = 0 r′−→ cc(K′)\ yJrJ−→ cc(K)

is a deformation retraction, and it satisfies ω(r(y)) = ω(y) as this is true for rJand r′. The inductive step is now complete. The required retraction r : U(K)→ ZK


covers r as shown in the following commutative diagram:

ZK

//

µ

U(K) r //

µ

ZK

µ

cc(K) // U>(K) r // cc(K)

Explicitly, r is decomposed inductively in a way similar to (4.23),

r : U(K) = U(K′)\LJ r′−→ ZK′\µ−1(yJ )rJ−→ ZK,

where µ−1(yJ) =∏j∈J0×

∏j /∈J S, and rJ is given in coordinates (z1, . . . , zm) =(√

y1eiϕ1 , . . . ,

√yme

iϕm)

by(√y1e

iϕ1 , . . . ,√yme

iϕm)7→(√y1 + α1y1e

iϕ1 , . . . ,√ym + αmyme

iϕm)

with αi as above.

Since U(K) and ZK are homotopy equivalent, we can use the results on thecohomology of ZK (such as Theorems 4.5.4 and 4.5.8) to describe the cohomologyrings of coordinate subspace arrangement complements. The additive isomorphismHk(U(K)) ∼=

⊕−i+2j=k Tor

−i,2jk[v1,...,vm](k[K],k) has been also proved in [124].

Example 4.7.6. Let K be the set of m disjoint points. Then ZK is homotopyequivalent to the complement U(K) of Example 4.7.4.3, and

Z[K] = Z[v1, . . . , vm]/(vivj , i 6= j).

The subspace of cocycles in R∗(K) has basis of monomials

ui1ui2 · · ·uik−1vik with ip 6= iq for p 6= q.

Since the total degree of ui1ui2 · · ·uik−1vik is k + 1, the space of cocycles of degree

k + 1 has dimension m(m−1k−1

). The subspace of coboundaries of degree k + 1 is

spanned by the elements of the form

d(ui1 · · ·uik)and has dimension

(mk

). Therefore,

rankH0(U(K)) = 1,

rankH1(U(K)) = H2(U(K)) = 0,

rankHk+1(U(K)) = m(m−1k−1

)−(mk

)= (k − 1)

(mk

), for 2 6 k 6 m,

and the multiplication in the cohomology of U(K) is trivial.

The calculation of the previous example shows that if K is the set of m points,then there is a cohomology ring isomorphism

(4.24) H∗(U(K)

) ∼= H∗( m∨

k=2

(Sk+1

)∨(k−1)(mk

)),

where X∨k denotes the k-fold wedge of a space X . This cohomology isomorphismis induced by a homotopy equivalence, as is shown by the following result.


Theorem 4.7.7 ([132], [133, Corollary 9.5]). Let K be the i-dimensional skele-ton of the simplex ∆m−1, so that U(K) is the complement to all coordinate planes ofcodimension i+2 in Cm. Then U(K) is homotopy equivalent to a wedge of spheres:

U(K) ≃m∨

k=i+2

(Si+k+1

)∨(mk)(k−1i+1).

The proof uses the homotopy fibration ZK → (CP∞)K → (CP∞)m (see The-orem 4.3.2); it will be further discussed in Section 8.2. For i = 0 we obtain thehomotopy equivalence behind cohomology isomorphism (4.24).

Real coordinate subspace arrangements

UR(K) = (R,R×)K

in Rm are defined and treated similarly; there are real analogues of all results andconstructions of this section. (The real version of Theorem 4.7.7 is much simpler:see Exercise 4.2.8).

A coordinate subspace can be given either by setting some coordinates to zero,as in (4.21), or as the linear span of a subset of the standard basis e1, . . . , em.The latter approach leads to an alternative way of parametrising complements ofcoordinate subspace arrangements by simplicial complexes, which is related to theformer one by Alexander duality.

Given a subset I ⊂ [m] we set SI = C〈e i : i ∈ I〉 (the C-span of the basis vectors

corresponding to I), and use the notation I = [m] \ I and K = I ⊂ [m] : I /∈ Kfrom Construction 2.4.1. Then the coordinate subspace arrangement correspondingto a simplicial complex K can be written in the following two ways:

A(K) = LI : I /∈ K = SI : I ∈ K.Using Alexander duality we can reformulate the description of the cohomology

of U(K) in terms of full subcomplexes of K (Theorem 4.5.8) as follows.

Proposition 4.7.8. There are isomorphisms

Hq(U(K)

) ∼=⊕

I∈K

H2m−2|I|−q−2(lkK I).

Proof. By Theorems 4.7.5 and 4.5.8,

Hq(U(K)

) ∼=⊕

I⊂[m]

Hq−|I|−1(KI).

Nonempty simplices I ∈ K do not contribute to the sum above, since the corre-

sponding subcomplexes KI are contractible. Since H−1(∅) = Z, the empty simplexcontributes Z to H0(U(K)). Therefore, we can rewrite the isomorphism above as

Hq(U(K)

)∼=⊕

I /∈K

Hq−|I|−1(KI).

Using Alexander duality (Proposition 2.4.6) we calculate

Hq−|I|−1(KI) ∼= H|I|−3−q+|I|+1(lkK I) = H2m−2|I|−q−2(lkK I),

where I = [m]\I is a simplex of K.


Proposition 4.7.8 is a particular case of the well-known Goresky–MacPhersonformula [130, Chapter III], which calculates the (co)homology groups of the com-plement of an arrangement of affine subspaces in terms of its intersection poset. Inthe case of coordinate subspace arrangements A(K) the intersection poset is the

face poset of the dual complex K. For more on the relationships between generalaffine subspace arrangements and moment-angle complexes see [55, Chapter 8].

Exercises.

4.7.9. The affine algebraic varietyX(K) corresponding to the face ring C[K] (seeProposition 3.1.12) and the coordinate subspace arrangement A(K) of Construc-

tion 4.7.1 are related by the identity X(K) = A(K), where K = I ⊂ [m] : [m] \ I /∈K is the Alexander dual complex.

4.7.10. Show directly that the complement to the 3 coordinate lines in C3 ishomotopy equivalent to the wedge of spheres S3∨S3∨S3∨S4∨S4; this correspondsto m = 3 in (4.24).

4.7.11. Show directly (without referring to Theorem 4.7.5 and Proposi-tion 4.3.5) that the complement U(K) is 2-connected if K does not have ghostvertices, and that U(K) is 2q-connected if K is q-neighbourly.

Moment-angle complexes: additional topics

4.8. Free and almost free torus actions on moment-angle complexes

Here consider free and almost free actions of toric subgroups T k ⊂ Tm on ZK.As usual, K is an (n − 1)-dimensional simplicial complex on [m], and ZK is thecorresponding moment-angle complex.

We start with a simple characterisation of the isotropy subgroups of the stan-dard Tm-action on ZK. For each I ⊂ [m] we consider the coordinate subtorus

TI = (t1, . . . , tm) ∈ Tm : tj = 1 for j /∈ I =∏

i∈I

T ⊂ Tm

(note that TI = (T, 1)I in the notation of Construction 4.2.1).

Proposition 4.8.1. Let z ∈ ZK, and set ω(z) = i ∈ [m] : zi = 0 ∈ K. Thenthe isotropy subgroup of z with respect to the Tm-action is Tω(z). Furthermore, eachcoordinate subtorus TI for I ∈ K is the isotropy subgroup for a point z ∈ ZK.

Proof. An element t = (t1, . . . , tm) ∈ Tm fixes z if and only if ti = 1 wheneverzi 6= 0, which is equivalent to that t ∈ Tω(z ). The last statement is also clear: TI

is the isotropy subgroup for any z ∈ (D, S)I ⊂ ZK with ω(z ) = I.

Recall that an action of a group on a topological space is almost free if allisotropy subgroups are finite.

Definition 4.8.2. We define the free toral rank of ZK, denoted ftrZK, as themaximal dimension of toric subgroups T k ⊂ Tm acting on ZK freely. Similarly, thealmost free toral rank of ZK, denoted atrZK, is the maximal dimension of toricsubgroups T k ⊂ Tm acting on ZK almost freely.

Proposition 4.8.3. Let K be a simplicial complex of dimension n − 1 on mvertices and K 6= ∆m−1. The toral ranks of ZK satisfy the following inequalities:

1 6 ftrZK 6 atrZK 6 m− n.Proof. By Proposition 4.8.1, isotropy subgroups for the Tm-action on ZK are

coordinate subgroups of the form TI . The diagonal circle in Tm intersects eachof these coordinate subgroups trivially (since I 6= [m]), and therefore acts freelyon ZK. This proves the first inequality. The second is obvious. To prove the thirdone, assume that T k ⊂ Tm acts almost freely on ZK. Then the intersection of T k

with every Tm-isotropy subgroup TI is a finite group. Choose a maximal simplexI ∈ K, |I| = n. Then TI ∩ T k can be finite only if k 6 m− n.

The map Rm → Tm, (ϕ1, . . . , ϕm) 7→ (e2πiϕ1 , . . . , e2πiϕm), identifies Tm withthe quotient Rm/Zm. Subtori T k ⊂ Tm of dimension k bijectively correspond tounimodular sublattices L ⊂ Zm of rank k (a sublattice is unimodular if it is a direct

169

170 MOMENT-ANGLE COMPLEXES: ADDITIONAL TOPICS

summand in Zm). The inclusion T k ⊂ Tm can be viewed as LR/L ⊂ Rm/Zm, whereLR is the k-dimensional subspace in Rm spanned by L.

Choosing a basis in L we obtain an integer m×k-matrix S = (sij), 1 6 i 6 m,1 6 j 6 k, so that L is identified with the image of S : Zk → Zm. The k-torus T k

is the image of the corresponding monomorphism of tori Tk → Tm, namely,

(4.25) T k =(e2πi(s11ψ1+···+s1kψk), . . . , e2πi(sm1ψ1+···+smkψk)

)⊂ Tm,

where (ψ1, . . . , ψk) ∈ Rk. Since L is unimodular, the columns of S form a part ofbasis of the lattice Zm.

Lemma 4.8.4. Let T k be a k-dimensional subtorus in Tm and let L be thecorresponding unimodular sublattice of rank k in Zm. Let S = (sij), 1 6 i 6 m,1 6 j 6 k, be a matrix defining L, so that T k is given by (4.25).

(a) The torus T k acts on ZK almost freely if and only if for each I ∈ Kthe intersection of subspaces LR and RI in Rm is zero. Equivalently, the(m−|I|)×k-matrix SI obtained by deleting from S the rows with numbersi ∈ I has rank k.

(b) The torus T k acts on ZK freely if and only if for each I ∈ K the sublatticespanned by L and ZI in Zm is unimodular of rank k + |I|. Equivalently,the columns of the (m− |I|)× k-matrix SI form a part of basis of Zm−|I|.

Proof. We prove (a) first. By Proposition 4.8.1, the T k-action on ZK isalmost free if and only if the intersection T k ∩ TI ⊂ Tm is finite for each I ∈ K.This intersection can be identified with the kernel of the map f : T k × TI → Tm

(the product of the inclusion maps T k → Tm and TI → Tm). This kernel is finiteif and only if the corresponding map of real spaces LR × RI → Rm is injective,which is equivalent to that LR ∩RI = 0. Let I = i1, . . . , ip, then the matrix off has the form (S |e i1 | · · · |eip), where ei is the ith standard basis column vector.Clearly, this matrix has rank k + |I| if and only if the matrix SI has rank k.

Now we prove (b). The T k-action on ZK is free if and only if the the kernel off : T k×TI → Tm is trivial for each I ∈ K, i.e. T k×TI embeds as a subtorus. Thisis equivalent to the conditions stated in (b).

Let t = (t1, . . . , tn) ∈ Z[K] be a linear sequence given by

(4.26) ti = λi1v1 + · · ·+ λimvm, for 1 6 i 6 n.

We consider the integer n×m-matrix Λ = (λij), 1 6 i 6 n, 1 6 j 6 m. It defines ahomomorphism of lattices Λ: Zm → Zn and a homomorphism of tori Λ: Tm → Tn.

Theorem 4.8.5. The following conditions are equivalent:

(a) the sequence (t1, . . . , tn) given by (4.26) is an lsop in the rational face ringQ[K];

(b) the kernel TΛ = Ker(Λ: Tm → Tn) is a product of an (m − n)-torus anda finite group, and TΛ acts almost freely on ZK.

4.8. FREE AND ALMOST FREE TORUS ACTIONS ON MOMENT-ANGLE COMPLEXES171

Proof. We first observe that under any of conditions (a) or (b) the rationalmap Λ: Qm → Qn is surjective. For each simplex I ∈ K we consider the diagram

0↓

Qm−n

yi1

0 −−−−→ Q|I| i2−−−−→ Qmp2−−−−→ Qm−|I| −−−−→ 0

yΛ

Qn

↓0

where i2 is the inclusion of the coordinate subspace QI → Qm. The map Λi2 isgiven by the n × |I| matrix ΛI = (λij), 1 6 i 6 n, j ∈ I. By Lemma 3.3.2, thesequence (t1, . . . , tn) is a rational lsop if and only if the rank of ΛI is |I| for eachI ∈ K. Hence condition (a) of the theorem is equivalent to the injectivity of themap Λi2 for each I ∈ K.

On the other hand, by Lemma 4.8.4 (a), the action of the (m− n)-torus TΛ onZK is almost free if and only if i1(Qm−n) ∩ QI = 0 for each I ∈ K. The lattercondition is equivalent to that i1(Qm−n) ∩Ker p2 = 0, i.e. that p2i1 is injective.Hence condition (b) of the theorem is equivalent to the injectivity of the map p2i1for each I ∈ K. Now the theorem follows from Lemma 1.2.5.

The almost free toral rank of ZK can now be easily determined:

Corollary 4.8.6. We have atrZK = m − n, i.e. for each simplicial complexK of dimension (n − 1) on [m] there is an (m − n)-dimensional subtorus in Tm

acting on ZK almost freely.

Proof. Choose a rational lsop in Q[K] by Theorem A.3.10, and multiply itby a common denominator to get an integral sequence (4.26). It is still an lsop inQ[K] (but it may fail to be an lsop in Z[K]), and therefore the (m − n)-torus TΛacts on ZK almost freely by Theorem 4.8.5.

There is an analogue of Theorem 4.8.5 for free torus actions:

Theorem 4.8.7. The following conditions are equivalent:

(a) the sequence (t1, . . . , tn) given by (4.26) is an lsop in Z[K];(b) TΛ = Ker(Λ: Tm → Tn) is an (m− n)-torus acting freely on ZK.

Proof. The argument is the same as in the proof of Theorem 4.8.5:Lemma 1.2.5 is now applied to the diagram of integral lattices instead of ratio-nal vector spaces.

Nevertheless, there is no analogue of Corollary 4.8.6 for free torus actions, asintegral lsop’s may fail to exist. Indeed, the free toral rank of ZK where K is theboundary of a cyclic n-polytope with m > 2n vertices is strictly less than m−n, asshown by Example 3.3.4. The free toral rank of ZK is a combinatorial characteristicof K, also known as the Buchstaber invariant . Its determination is a very subtleproblem; see [107] and [120] for some partial results in this direction.


There is the following important conjecture of equivariant topology and rationalhomotopy theory concerning almost free torus action.

Conjecture 4.8.8 (Toral Rank Conjecture, Halperin [148]). Assume that atorus T k acts almost freely on a finite-dimensional topological space X. Then

dimH∗(X ;Q) > 2k,

i.e. the total dimension of the cohomology of X is at least that of the torus T k.

The Toral Rank Conjecture is valid for k 6 3 and is open in general. See [269]and [312] for the discussion of the current status of this conjecture.

In the case of moment-angle complexes we have the following result:

Theorem 4.8.9 ([66], [311]). Let K be a simplicial complex of dimension n−1with m vertices, and let ZK be the corresponding moment-angle complex. Then

rankH∗(ZK) =

m+n∑

k=0

rankHk(ZK) > 2m−n.

The proof of this theorem given in [311] uses a construction of independentinterest and a couple of technical lemmata. We include this proof below.

Corollary 4.8.10. The Toral Rank Conjecture is valid for subtori T k ⊂ Tm

acting almost freely on ZK.

The following is a particular case of the so-called simplicial wedge construc-tion [268]. It has been brought into toric topology by the work [15].

Construction 4.8.11 (Simplicial doubling). Let K be a simplicial complexon the vertex set [m]. The double of K is the simplicial complex D(K) on thevertex set [2m] = 1, 1′, 2, 2′, . . . ,m,m′ whose missing faces (minimal non-faces)are i1, i′1, . . . , ik, i′k where i1, . . . , ik is a missing face of K. In other words, D(K)is determined by its face ring given by

k[D(K)

]= k[v1, v1′ , . . . , vm, vm′ ]

/(vi1vi′1 · · · vikvi′k : i1, . . . , ik /∈ K

).

Example 4.8.12.1. If K = ∆m−1 (the full simplex on m vertices), then D(K) = ∆2m−1.2. If K = ∂∆m−1, then D(K) = ∂∆2m−1.

The doubling construction interacts nicely with the polyhedral product:

Theorem 4.8.13. Let (X,A) be a pair of spaces, let K be a simplicial complexon [m] and D(K) its double. Then

(X,A)D(K) = (X ×X,X ×A ∪ A×X)K.

Proof. Set (Y,B) = (X×X,X×A∪A×X). Given a point y = (y1, . . . , ym) ∈Y m, we set

ωY (y) = i ∈ [m] : yi /∈ B ⊂ [m].

Similarly, given x = (x1, x1′ , . . . , xm, xm′) ∈ X2m, we set

ωX(x ) =j ∈ 1, 1′, . . . ,m,m′ : xj /∈ A

⊂ 1, 1′, . . . ,m,m′.

We identify y with x by the formula (y1, . . . , ym) = ((x1, x1′), . . . , (xm, xm′)) ∈Y m = X2m. It follows from the definition of the polyhedral product that y /∈

4.8. FREE AND ALMOST FREE TORUS ACTIONS ON MOMENT-ANGLE COMPLEXES173

(Y,B)K if and only if ωY (y) /∈ K. The latter is equivalent to the condition ωX(x ) /∈D(K), since if ωY (y) = i1, . . . , ik then ωX(x ) ⊃ i1, i′1, . . . , ik, i′k. Therefore,

y /∈ (Y,B)K ⇐⇒ x /∈ (X,A)D(K),

which implies that (X,A)D(K) = (Y,B)K.

Remark. The simplicial wedge [268], [15] is a generalisation of the doublingconstruction, in which the ith vertex of K is replaced by a ji-tuple of vertices,for some vector j = (j1, . . . , jm) of natural numbers. The double corresponds toj = (2, . . . , 2). There is an analogue of Theorem 4.8.13 in this setting, see [15, §7].

As an important consequence of Theorem 4.8.13 we obtain the following rela-tionship between the moment-angle complex ZK and its real analogue RK:

Corollary 4.8.14. We have ZK∼= RD(K).

Proof. Apply Theorem 4.8.13 to the pair (X,A) = (D1, S0), observing that(D1 ×D1, D1 × S0 ∪ S0 ×D1) ∼= (D2, S1).

Lemma 4.8.15. Let (X,A) be a pair of cell complexes such that A has a collarneighbourhood U(A) in X (i.e. there is a homeomorphism of pairs (U(A), A) ∼=(A × [0, 1), A × 0)). Let Y = X1 ∪A X2 be the space obtained by attaching twocopies of X along A. Then

rankH∗(Y ) > rankH∗(A).

Proof. The assumption on (X,A) implies that we can apply the Mayer–Vietoris sequence to the decomposition Y = X1 ∪A X2:

· · · βk−1−→ Hk−1(A)δk−1−→ Hk(Y )

αk−→ Hk(X1)⊕Hk(X2)βk−→ Hk(A)→ · · · .

The map βk is i∗1 ⊕ (−i∗2), where i1 : A → X1 and i2 : A → X2 are the inclusions.Since X1 = X2 = X , and the inclusions i1 and i2 coincide, we have rankKerβk >

rankHk(X) and rank Imβk 6 rankHk(X). Using these inequalities we calculate

rankHk(Y ) = rankKerαk + rank Imαk = rank Im δk−1 + rankKerβk

> rankHk−1(A)− rank Imβk−1 + rankHk(X)

> rankHk−1(A)− rankHk−1(X) + rankHk(X).

The required inequality is obtained by summing up over k.

Theorem 4.8.16. Let K be a simplicial complex of dimension n − 1 with mvertices, and let RK be the corresponding real moment-angle complex. Then

rankH∗(RK) > 2m−n′

> 2m−n,

where n′ is the minimum of the cardinality of maximal simplices of K (so thatn′ = n = dimK + 1 if and only if K is pure).

Proofs of Theorems 4.8.9 and 4.8.16. We first prove the inequality forRK, by induction on the number of vertices m. For m = 1 the statement is clear.We embed RK as a subcomplex in the ‘big’ cube [−1, 1]m (see Construction 4.1.5)with coordinates u = (u1, . . . , um), −1 6 ui 6 1. Assume that the first vertex


of K belongs to an (n′ − 1)-dimensional maximal simplex of K, and consider thefollowing subspaces of RK:

X+ = u ∈ RK : u1 > 0, X− = u ∈ RK : u1 6 0,A = X+ ∩X− = u ∈ RK : u1 = 0.

Applying Lemma 4.8.15 to the decomposition RK = X+ ∪A X− we obtain

rankH∗(RK) > rankH∗(A).

On the other hand, A is the disjoint union of 2m−m1−1 copies of RlkK1, where m1

is the number of vertices of lkK1. Since 1 is a vertex of a maximal simplex of Kof minimal cardinality n′, the minimal cardinality of maximal simplices in lkK1is n′

1 = n′ − 1. Now using the inductive hypothesis we obtain

rankH∗(A) = 2m−m1−1 rankH∗(RlkK1) > 2m−m1−12m1−n′1 = 2m−n′

.

Theorem 4.8.16 is therefore proved.To prove Theorem 4.8.9 we use the fact that ZK

∼= RD(K), and observe thatthe numbers m− n (and m− n′) for K and D(K) coincide.

Using Theorems 4.5.4 and 4.5.8 we may reformulate Theorem 4.8.9 in bothalgebraic and combinatorial terms:

Theorem 4.8.17. Let K be a simplicial complex of dimension n − 1 with mvertices, and let k be a field. Then

dimTork[v1,...,vm]

(k[K],k

)=

∑

J⊂[m], k>0

dim Hk−|J|−1(KJ ;k) > 2m−n.

As a corollary we obtain that the weak Horrocks Conjecture (Conjecture A.2.12)holds for a particular class of rings:

Corollary 4.8.18. Let K be a Cohen–Macaulay simplicial complex (over afield k) of dimension n− 1 with m vertices. Let t = (t1, . . . , tn) be an lsop in k[K],so that k[m]/t ∼= k[w1, . . . , wm−n] and dimk[K]/t <∞. Then

dimTork[w1,...,wm−n]

(k[K]/t ,k

)> 2m−n,

i.e. the weak Horrocks Conjecture holds for the rings k[K]/t.Proof. This follows from the previous theorem and Proposition 4.5.6.

Exercises.

4.8.19. Show that ftrZK = 1 if and only if K = ∂∆m−1.

4.8.20. The Toral Rank Conjecture fails if dimX =∞.

4.8.21. Show that the doubling operation respects the join, that is, D(K∗L) =D(K) ∗ D(L).

4.8.22. Assume that K is the boundary complex of a simplicial n-polytope Q ⊂Rn with m vertices v1, . . . , vm. Then D(K) is the boundary of a simplicial polytopeD(Q) of dimension m+n with 2m vertices, which can be obtained in the followingway. We embed Rn as the coordinate subspace in Rm+n on the last n coordinates.For each vertex vi ∈ Q ⊂ Rn take the line li ⊂ Rm+n through vi parallel to the ithcoordinate line of Rm+n, for 1 6 i 6 m. Then replace each vi by a pair of points

4.9. MASSEY PRODUCTS IN THE COHOMOLOGY OF MOMENT-ANGLE COMPLEXES175

v′i, v′′i ∈ li such that vi the centre of the segment with the vertices v′i, v

′′i . Then the

boundary ofD(Q) = conv(v′1, v

′′1 , . . . , v

′m, v

′′m) ⊂ Rm+n

is D(K).4.8.23. There is the following generalisation of Theorem 4.8.13. Let

(X ,A) =(X1, A1), (X1′ , A1′), . . . , (Xm, Am), (Xm′ , Am′)

be a set of 2m pairs of spaces. Define a new set (Y ,B) = (Y1, B1), . . . , (Ym, Bm)of m pairs, where

(Yi, Bi) =(Xi ×Xi′ , Xi ×Ai′ ∪ Ai ×Xi′

).

Then(X ,A)D(K) ∼= (Y ,B)K.

Further generalisations can be found in [15, §7].

4.8.24. The inequality rankH∗(RK) > 2m−n of Theorem 4.8.16 (or the in-equality rankH∗(ZK) > 2m−n) turns into identity if and only if

K = ∂∆k1−1 ∗ ∂∆k2−1 ∗ · · · ∗ ∂∆kp−1 ∗∆m−s−1,

where s = k1 + · · · + kp and the join factor ∆m−s−1 is void if s = m (compareExercise 3.3.19). In this case both RK and ZK are products of spheres and a disc.

4.9. Massey products in the cohomology of moment-angle complexes

Here we address the question of existence of nontrivial triple Massey productsin the Koszul complex

(Λ[u1, . . . , um]⊗ Z[K], d)of the face ring, and therefore in the cohomology of ZK. The general definition ofMassey products in the cohomology of a differential graded algebra is reviewed inSection A.4 of the Appendix. A geometrical approach to constructing nontrivialtriple Massey products in the Koszul complex of the face ring was developed byBaskakov in [23] as an extension of the cohomology calculation in Theorem 4.5.7.It is well-known that non-trivial higher Massey products obstruct the formalityof a differential graded algebra, which in our case leads to a family of nonformalmoment-angle manifolds ZK (see Section B.2 for the background material).

Construction 4.9.1 (Baskakov [23]). Let Ki be a triangulation of a sphereSni−1 with |Vi| = mi vertices, i = 1, 2, 3. Set m = m1+m2+m3, n = n1+n2+n3,

K = K1 ∗ K2 ∗ K3, ZK = ZK1 ×ZK2 ×ZK3 .

Then K is a triangulation of Sn−1 and therefore ZK is an (m+ n)-manifold.We choose maximal simplices I1 ∈ K1, I ′2, I

′′2 ∈ K2 such that I ′2 ∩ I ′′2 = ∅, and

I3 ∈ K3. SetK = ssI1∪I′2(ssI′′2 ∪I3 K),

where ssI denotes the stellar subdivision at I, see Definition 2.7.1. Then K is atriangulation of Sn−1 with m+ 2 vertices. Take generators

βi ∈ Hni−1(KVi) ∼= Hni−1(Sni−1), for i = 1, 2, 3,

where KViis the restriction of K to the vertex set of Ki, and set

αi = h(βi) ∈ Hni−mi,2mi(ZK) ⊂ Hmi+ni(ZK),


where h is the isomorphism of Theorem 4.5.7. Then

β1β2 ∈ Hn1+n2−1(KV1⊔V2)∼= Hn1+n2−1(Sn1+n2−1 \ pt = 0,

hence α1α2 = h(β1β2) = 0, and similarly α2α3 = 0. Therefore, the triple Masseyproduct 〈α1, α2, α3〉 ⊂ Hm+n−1(ZK) is defined. By definition, it is the set of

cohomology classes represented by the cocycles (−1)deg a1+1a1f + ea3 where ai is acocycle representing αi, and e, f are cochains satisfying de = a1a2, df = a2a3.

For the simplest example of this series, take Ki = S0 (two points), so that K is

the boundary of an octahedron, and K is obtained by applying stellar subdivisionsat two skew edges. We shall consider this example in more detail below.

Recall that a Massey product is trivial if it contains zero.

Theorem 4.9.2. The above defined triple Massey product

〈α1, α2, α3〉 ⊂ Hm+n−1(ZK)

is nontrivial.

Proof. Consider the subcomplex of K consisting of the two new vertices addedto K in the process of stellar subdivision. By Proposition 4.2.3 (b), the inclusion ofthis subcomplex induces an embedding of a 3-dimensional sphere S3 → ZK . Sincethe two new vertices are not joined by an edge in ZK, the embedded 3-sphere defines

a non-trivial class x ∈ H3(ZK). Its Poincare dual cohomology class in Hm+n−1(ZK)is contained in the Massey product 〈α1, α2, α3〉. We need only to check that thiselement cannot be turned into zero by adding elements from the indeterminacy ofthe Massey product, i.e. from the subspace

α1 ·Hm2+m3+n2+n3−1(ZK) + α3 ·Hm1+m2+n1+n2−1(ZK).

To do this we use the multigraded structure in H∗(ZK). The multigraded com-

ponents of the group Hm2+m3+n2+n3−1(ZK) different from the component deter-

mined by the full subcomplex KV2⊔V3 do not affect the nontriviality of the Massey

product, while the multigraded component corresponding to KV2⊔V3 is zero since

KV2⊔V3∼= Sn2+n3−1 \ pt is contractible. The group Hm1+m2+n1+n2−1(ZK) is con-

sidered similarly. It follows that the Massey product contains a unique nonzeroelement in its multigraded component and therefore it is nontrivial.

Corollary 4.9.3. Let K be a triangulated sphere obtained from another sphereK by applying two stellar subdivisions as described above. Then the corresponding2-connected moment-angle manifold ZK is non-formal.

In the proof of Theorem 4.9.2 the nontriviality of the Massey product is estab-lished geometrically. A parallel argument may be carried out algebraically using theKoszul complex or its quotient algebra R∗(K), as illustrated in the next example.To be precise, the nonformality of a manifold ZK is equivalent to the nonformalityof its singular cochain algebra C∗(ZK;Q) (or Sullivan’s algebra APL(ZK)), whilethe rational Koszul complex or the algebra R∗(K)⊗Q are quasi-isomorphic to thecellular cochain algebra C∗(ZK;Q). However, this difference is irrelevant: it can beeasily seen that the existence of a nontrivial triple Massey product for the cellularcochain algebra C∗(ZK;Q) implies that ZK is nonformal (an exercise, or see [95,Proposition 5.6.1]).

4.9. MASSEY PRODUCTS IN THE COHOMOLOGY OF MOMENT-ANGLE COMPLEXES177

Example 4.9.4. We consider the simplest case of Construction 4.9.1, when Kis obtained from the boundary of an octahedron by applying stellar subdivisions

at two skew edges. Then K = KP is the nerve complex of a simple 3-polytope Pobtained by truncating a cube at two edges as shown in Figure 4.3. The face ringis given by

Z[KP ] = Z[v1, . . . , v6, w1, w2]/IKP,

where vi, i = 1, . . . , 6, are the generators coming from the facets of the cube andw1, w2 are the generators corresponding to the two new facets, and

IKP= (v1v2, v3v4, v5v6, w1w2, v1v3, v4v5, w1v3, w1v6, w2v2, w2v4).

We denote the corresponding exterior generators of R∗(KP ) by u1, . . . , u6, t1, t2;

v1

v2

v3

v4

v5

v6

w1

w2

Figure 4.3.

they satisfy dui = vi and dti = wi. Consider the cocycles

a = v1u2, b = v3u4, c = v5u6

and the corresponding cohomology classes α, β, γ ∈ H−1,4[R∗(K)]. The equations

ab = de, bc = df

have a solution e = 0, f = v5u3u4u6, so the triple Massey product 〈α, β, γ〉 ∈H−4,12[R∗(K)] is defined. This Massey product is represented by the cocycle

af + ec = v1v5u2u3u4u6

and is nontrivial. The differential graded algebra R∗(KP ) and the 11-dimensionalmanifold ZKP

are not formal.

Remark. We can truncate the polytope P from the previous example at an-other edge to obtain a 3-dimensional associahedron As3, shown in Figure 1.5 (left).By considering similar nontrivial Massey products (now there will be three of them,corresponding to each pair of cut off edges) we deduce that the 12-dimensionalmoment-angle manifold corresponding to As3 is also nonformal.

In view of Theorem 4.9.2, the question arises of describing the class of sim-plicial complexes K for which the algebra R∗(K) (equivalently, the Koszul algebra(Λ[u1, . . . , um]⊗Z[K], d)) is formal. For example, this is the case if K is the bound-ary of a polygon or, more generally, if K is of the form described in Theorem 4.6.12.


Triple Massey products in the cohomology of ZK were further studied in thework of Denham and Suciu [95]. According to [95, Theorem 6.1.1], there existsa nontrivial triple Massey product of 3-dimensional cohomology classes α, β, γ ∈H3(ZK) if and only if the 1-skeleton of K contains an induced subgraph isomorphicto one of the five explicitly described ‘obstruction’ graphs. In [95, Example 8.5.1]there is also constructed an example of K for which the corresponding ZK has anindecomposable triple Massey product in the cohomology (a triple Massey productis indecomposable if it does not contain a cohomology class that can be written asa product of two cohomology classes of positive dimension).

To conclude this section, we mention that the algebraic study of Massey prod-ucts in the cohomology of Koszul complexes has a long history. It goes back to thework of Golod [128], who studied the Poincare series of TorR(k,k) for a Notherianlocal ring R. The main result of Golod is a calculation of the Poincare seriesfor the class of rings whose Koszul complexes have all Massey products vanish-ing (including the cohomology multiplication). Such rings were called Golod inthe monograph [147] of Gulliksen and Levin, where the reader can find a detailedexposition of Golod’s theorem together with several further applications.

Definition 4.9.5. We refer to a simplicial complex K as Golod (over a ring k)if its face ring k[K] has the Golod property, i.e. if the multiplication and all higherMassey products in Tork[v1,...,vm](k[K],k) = H(Λ[u1, . . . , um]⊗ k[K], d) are trivial.

Golod complexes were studied in [157], where several combinatorial criteria forGolodness were given. The appearance of moment-angle complexes added a topo-logical dimension to the whole study. In particular, Theorem 4.5.4 implies that K isGolod whenever ZK is homotopy equivalent to a wedge of spheres. This observationwas used in [133, Theorems 9.1, 11.2] to produce new classes of Golod simplicialcomplexes, including skeleta of simplices considered in Theorem 4.7.7, and, moregenerally, all shifted complexes. For all such K the corresponding moment-anglecomplex ZK is homotopy equivalent to a wedge of spheres. There are examplesof Golod complexes K for which ZK is not homotopy equivalent to a wedge ofspheres (see Exercise 4.9.7 and [131, Example 3.3]). More explicit series of Golodcomplexes were constructed by Seyed Fakhari and Welker in [284].

By a result of Berglund and Jollenbek [28, Theorem 5.1], the face ring k[K]is Golod if and only if the multiplication in Tork[v1,...,vm](k[K],k) is trivial (i.e.triviality of the cup-product implies that all higher Massey products are also trivial).

More details on the relationship between the Golod property for K and thehomotopy theory of moment-angle complexes ZK can be found in [133], [131], aswell as in Section 8.5.

Exercises.

4.9.6. If the cellular cochain algebra C∗(ZK;Q) carries a nontrivial triple Masseyproduct, then ZK is nonformal.

4.9.7. Let K be the triangulation of RP 2 from Example 3.2.12.4. Then K is aGolod complex, but ZK is not homotopy equivalent to a wedge of spheres.

4.10. Moment-angle complexes from simplicial posets

Simplicial posets S generalise naturally abstract simplicial complexes (see Sec-tion 2.8). Algebraic properties of their face rings k[S] were discussed in Section 3.5.

4.10. MOMENT-ANGLE COMPLEXES FROM SIMPLICIAL POSETS 179

Following the categorical description of the moment-angle complex ZK outlined inthe end of Construction 4.1.1, it is easy to extend the definition of ZK to simpli-cial posets. The resulting space ZS carries a torus action, and its equivariant andordinary cohomology is expressed in terms of the face ring Z[S] in the same wayas for the standard moment-angle complexes ZK. Simplicial posets and associatedmoment-angle complexes therefore provide a broader context for studying the linkbetween torus actions and combinatorial commutative algebra. These developmentsare originally due to Lu and Panov [199].

Let S be a finite simplicial poset with the vertex set V (S) = [m].

Construction 4.10.1 (moment-angle complex). We consider the face categorycat(S) whose objects are elements σ ∈ S and there is a morphism from σ to τwhenever σ 6 τ . For each element σ ∈ S we define the following subset in thestandard unit polydisc Dm ⊂ Cm:

(D2, S1)σ =(z1, . . . , zm) ∈ Dm : |zj |2 = 1 for j 66 σ

.

Then (D2, S1)σ is homeomorphic to a product of |σ| discs and m− |σ| circles. Wehave an inclusion (D2, S1)τ ⊂ (D2, S1)σ whenever τ 6 σ. Now define a diagram

DS(D2, S1) : cat(S) −→ top,

σ 7−→ (D2, S1)σ,

which maps a morphism σ 6 τ of cat(S) to the inclusion (D2, S1)σ ⊂ (D2, S1)τ

(see Appendix C.1 for the definition of diagrams and their colimits).We define the moment-angle complex corresponding to S by

ZS = colimDS(D2, S1) = colim

σ∈S(D2, S1)σ.

The space ZS is therefore glued from the blocks (D2, S1)σ according to the posetrelation in S. When S is (the face poset of) a simplicial complex K it becomes thestandard moment-angle complex ZK.

Since every subset (D2, S1)σ ⊂ Dm is invariant with respect to the coordinate-wise action of the m-torus Tm, the moment-angle complex ZS acquires a Tm-action.

This definition extends to a set (X ,A) of m pairs of spaces (see Construc-tion 4.2.1), so we define the polyhedral product of (X ,A) corresponding to S by

(X ,A)S = colimDS(X ,A) = colimσ∈S

(X ,A)σ.

The construction of the polyhedral product (X ,A)S is functorial in all argu-ments: there are straightforward analogues of Propositions 4.2.3 and 4.2.4, whichare proved in a similar way.

Example 4.10.2. Let S be the simplicial poset of Fig. 3.3 (a). Then ZS isobtained by gluing two copies of D2 × D2 along their boundary S3 = D2 × S1 ∪S1×D2. Therefore, ZS

∼= S4. Here, KS = ∆1 (a segment), and the moment-anglecomplex map induced by the map S → KS (2.9) folds S4 onto D4. Similarly, if Sis of Fig. 3.3 (b), then ZS

∼= S6. Note that even-dimensional spheres do not appearas moment-angle complexes ZK for simplicial complexes K.

The join of simplicial posets S1 and S2 is the simplicial poset S1 ∗ S2 whoseelements are pairs (σ1, σ2), with (σ1, σ2) 6 (τ1, τ2) whenever σ1 6 τ1 in S1 andσ2 6 τ2 in S2. The following properties of ZS are similar to those of ZK.


Theorem 4.10.3.

(a) ZS1∗S2∼= ZS1 ×ZS2 ;

(b) the quotient ZS/Tm is homeomorphic to the cone over |S|;(c) if |S| ∼= Sn−1, then ZS is a manifold of dimension m+ n.

Proof. Statements (a) and (b) are proved in the same way as the correspond-ing statements for simplicial complexes, see Section 4.1. To prove (c) we use the‘dual’ decomposition of the boundary of the n-ball cone |S| into faces, in the sameway as in the proof of Theorem 4.1.4.

Construction 4.10.4 (cell decomposition). We proceed by analogy with theconstruction of Section 4.4. The disc D is decomposed into 3 cells: the point 1 ∈ Dis the 0-cell; the complement to 1 in the boundary circle is the 1-cell, which wedenote T ; and the interior of D is the 2-cell, which we denote D. The polydiscDm then acquires the product cell decomposition, with each (D2, S1)σ ⊂ (D2, S1)τ

being an inclusion of cellular subcomplexes for σ 6 τ . We therefore obtain a celldecomposition of ZS (although this time it is not a subcomplex in Dm in general!).Each cell in ZS is determined by an element σ ∈ S and a subset ω ∈ V (S) withV (σ) ∩ ω = ∅. Such a cell is a product of |σ| cells of D-type, |ω| cells of T -typeand the rest of 1-type. We denote this cell by κ(ω, σ).

The resulting cellular cochain complex C∗(ZS) has an additive basis consistingof cochains κ(ω, σ)∗ dual to the corresponding cells. We introduce a Z⊕Zm-gradingon the cochains by setting

mdeg κ(ω, σ)∗ = (−|ω|, 2V (σ) + 2ω),

where we think of both V (σ) and ω as vectors in 0, 1m ⊂ Zm. The cellulardifferential preserves the Zm-part of the multigrading, so we obtain a decomposition

C∗(ZS) =⊕

a∈Zm

C∗,2a (ZK)

into a sum of subcomplexes. The only nontrivial subcomplexes are those for whicha is in 0, 1m. The cellular cohomology of ZS thereby acquires a multigrading,and we define the multigraded Betti numbers b−i,2a (ZS) by

b−i,2a (ZS) = rankH−i,2a (ZS), for 1 6 i 6 m, a ∈ Zm.

For the ordinary Betti numbers we have bk(ZS) =∑

−i+2|a |=k b−i,2a (ZS).

The map of moment-angle complexes ZS1 → ZS2 induced by a simplicial posetmap S1 → S2 is clearly a cellular map, and therefore the cellular cohomology isfunctorial with respect to such maps.

We now recall from Section 3.5 that the face ring Z[S] is a Zm-gradedZ[v1, . . . , vm]-module via the map sending each vi identically, and we have theZ⊕ Zm-graded Tor-algebra of Z[S]:

TorZ[v1,...,vm]

(Z[S],Z

)=

⊕

i>0,a∈Nm

Tor−i, 2aZ[v1,...,vm]

(Z[S],Z

).

Theorem 4.10.5. There is a graded ring isomorphism

H∗(ZS) ∼= TorZ[v1,...,vm](Z[S],Z)


whose graded components are given by the group isomorphisms

(4.27) Hp(ZS) ∼=⊕

−i+2|a|=p

Tor−i, 2aZ[v1,...,vm](Z[S],Z)

in each degree p. Here |a| = j1 + · · ·+ jm for a = (j1, . . . , jm).

Using the Koszul complex we restate the above theorem as follows:

Theorem 4.10.6. There is a graded ring isomorphism

H∗(ZS) ∼= H(Λ[u1, . . . , um]⊗ Z[S], d

),

where the Z⊕Zm-grading and the differential on the right hand side are defined by

mdeg ui = (−1, 2ei), mdeg vσ = (0, 2V (σ)), dui = vi, dvσ = 0,

and ei ∈ Zm is the ith basis vector, for i = 1, . . . ,m.

Proof. The proof follows the lines of the proof of Theorem 4.5.4, but theanalogues of Lemmata 3.2.6 and 4.5.3 are proved in a different way.

We first set up the quotient differential graded ring

R∗(S) = Λ[u1, . . . , um]⊗ Z[S]/IRwhere IR is the ideal generated by the elements

uivσ with i ∈ V (σ), and vσvτ with σ ∧ τ 6= 0.

Note that the latter condition is equivalent to V (σ) ∩ V (τ) 6= ∅. The ring R∗(S)will serve as an algebraic model for the cellular cochains of ZS .

We need to prove an analogue of Lemma 3.2.6, i.e. show that the quotient map

: Λ[u1, . . . , um]⊗ Z[S]→ R∗(S)induces an isomorphism in cohomology. Instead of constructing a chain homotopydirectly, we shall identify both R∗(S) and Λ[u1, . . . , um] ⊗ Z[S] with the cellularcochains of homotopy equivalent spaces.

Theorem 3.5.7 implies that R∗(S) has basis of monomials uωvσ where ω ⊂V (S), σ ∈ S, ω ∩ V (σ) = ∅, and uω = ui1 . . . uik for ω = i1, . . . , ik. Inparticular, R∗(S) is a free abelian group of finite rank. The map

(4.28) g : R∗(S)→ C∗(ZS), uωvσ 7→ κ(ω, σ)∗

is an isomorphism of cochain complexes. Indeed, the additive bases of the twogroups are in one-to-one correspondence, and the differential in R∗(S) acts (in thecase |ω| = 1 and i /∈ V (σ)) as

d(uivσ) = vivσ =∑

η∈i∨σ

vη.

This is exactly how the cellular differential in C∗(ZS) acts on κ(i, σ)∗. The caseof arbitrary ω is treated similarly. It follows that we have an isomorphism ofcohomology groups Hj [R∗(S)] ∼= Hj(ZS) for all j.

The differential algebra (Λ[u1, . . . , um] ⊗ Z[S], d) also may be identified withthe cellular cochains of a certain space. Namely, consider the polyhedral product(S∞, S1)S . Then we show in the same way as in Subsection 4.5 that there is anisomorphism of cochain complexes

g′ : Λ[u1, . . . , um]⊗ Z[S]→ C∗((S∞, S1)S

).


Furthermore, the standard functoriality arguments give a deformation retraction

ZS = (D2, S1)S → (S∞, S1)S −→ (D2, S1)S

onto a cellular subcomplex. Hence the cochain map C∗((S∞, S1)S)→ C∗(ZS) corre-sponding to the inclusion ZS → (S∞, S1)S induces an isomorphism in cohomology.

Summarising the above observations we obtain the commutative square

(4.29)

Λ[u1, . . . , um]⊗ Z[S] g′−−−−→ C∗((S∞, S1)S)

yy

R∗(S) g−−−−→ C∗(ZS)

in which the horizontal arrows are isomorphisms of cochain complexes, and theright vertical arrow induces an isomorphism in cohomology. It follows that the leftarrow also induces an isomorphism in cohomology, as claimed.

The additive isomorphism of (4.27) now follows from (4.29). To establish thering isomorphism we need to analyse the multiplication of cellular cochains.

We consider the diagonal approximation map ∆ : Dm → Dm × Dm given oneach coordinate by (4.15). It restricts to a map (D2, S1)σ → (D2, S1)σ × (D2, S1)σ

for every σ ∈ S and gives rise to a map of diagrams

DS(D2, S1)→ DS(D

2, S1)×DS(D2, S1).

By definition, the colimit of the latter is ZS ∗S , which is identified with ZS × ZS .

We therefore obtain a cellular approximation ∆ : ZS → ZS × ZS for the diagonalmap of ZS . It induces a ring structure on the cellular cochains via the composition

C∗(ZS)⊗ C∗(ZS)×−−−−→ C∗(ZS ×ZS)

∆∗

−−−−→ C∗(ZS).

We claim that, with this multiplication in C∗(ZS), the map (4.28) becomes a ringisomorphism. To see this we first observe that since (4.28) is a linear map, it isenough to consider the product of two generators uωvσ and uψvτ . If any two of thesubsets ω, V (σ), ψ and V (τ) have nonempty intersection, then uωvσ · uψvτ = 0 inR∗(S). Otherwise (if all four subsets are disjoint) we have

(4.30) g(uωvσ · uψvτ ) = g(uω⊔ψ ·

∑

η∈ σ∨τ

vη)=

∑

η∈σ∨τ

κ(ω ⊔ ψ, η)∗.

We also observe that for any cell κ(χ, η) of ZS (with χ ∩ V (η) = ∅) we have

∆κ(χ, η) =∑

ω⊔ψ=χσ∨τ ∋ η

κ(ω, σ)× κ(ψ, τ).

Therefore,

g(uωvσ) · g(uψvτ ) = κ(ω, σ)∗ · κ(ψ, τ)∗

= ∆∗(κ(ω, σ)× κ(ψ, τ)

)∗=

∑

η∈σ∨τ

κ(ω ⊔ ψ, η)∗.

Comparing this with (4.30) we deduce that (4.28) is a ring map.

Remark. Using the monoid structure on D as in Proposition 4.2.4 one easilysees that the construction of ZS is functorial with respect to maps of simplicialposets. This together with Proposition 3.5.12 makes the isomorphism of Theo-rem 4.10.5 functorial.


SF2

e

F4

F3F5

4

2

5

1

g

3F1

σ

f

Q

Figure 4.4. ‘Ball with corners’ Q dual to the simplicial poset S.

Using Hochster’s formula for simplicial posets (Theorem 3.5.14) we can calcu-late the cohomology of ZS via the cohomology of full subposets SJ ⊂ S. Here isan example of calculation using this method.

Example 4.10.7. Let S be the simplicial poset shown on Fig. 4.4 (right). Ithas m = 5 vertices, 9 edges and 6 triangular faces. The dual 3-dimensional ‘ballwith corners’ Q (see the proof of Theorem 4.10.3) is shown in Figure 4.4 (left). Wedenote its facets F1, . . . , F5, edges e, f, g and the vertex σ = F3 ∩ f as shown. Thecorresponding moment-angle complex ZS is an 8-dimensional manifold.

The face ring Z[S] is the quotient of the polynomial ring

Z[S] = Z[v1, . . . , v5, ve, vf , vg], deg vi = 2, deg ve = deg vf = deg ve = 4

by the relations

v1v2 = ve + vf + vg,

v3v4 = v3v5 = v4v5 = v3ve = v4vf = v5vg = vevf = vevg = vevf = 0.

The other generators and relations in the original presentation of Z[S] can be de-rived from the above; e.g., vσ = v3vf .

Given J ⊂ [m] we define the following subset in the boundary of Q:

QJ =⋃

j∈J

Fj ⊂ Q.

By analogy with Proposition 3.2.11 we prove that

(4.31) H−i, 2J (ZS) ∼= H |J|−i−1(QJ ).


Using this formula we calculate the nontrivial cohomology groups of ZS as follows:

H0,(0,0,0,0,0)(ZS) = H−1(∅) = Z 1

H−1,(0,0,2,2,0)(ZS) = H0(F3 ∪ F4) = Z u3v4

H−1,(0,0,2,0,2)(ZS) = H0(F3 ∪ F5) = Z u5v3

H−1,(0,0,0,2,2)(ZS) = H0(F4 ∪ F5) = Z u4v5

H−2,(0,0,2,2,2)(ZS) = H0(F3 ∪ F4 ∪ F5) = Z⊕ Z u5u3v4, u5u4v3

H0,(2,2,0,0,0)(ZS) = H1(F1 ∪ F2) = Z⊕ Z ve, vf

H−1,(2,2,2,0,0)(ZS) = H1(F1 ∪ F2 ∪ F3) = Z u3ve

H−1,(2,2,0,2,0)(ZS) = H1(F1 ∪ F2 ∪ F4) = Z u4vf

H−1,(2,2,0,0,2)(ZS) = H1(F1 ∪ F2 ∪ F5) = Z u5vg

H−2,(2,2,2,2,2)(ZS) = H2(F1 ∪ · · · ∪ F5) = Z u5u4v3vf = u5u4vσ

It follows that the ordinary (1-graded) Betti numbers of ZS are given by the se-quence (1, 0, 0, 3, 4, 3, 0, 0, 1). In the right column of the table above we includethe cocycles in the differential graded ring Λ[u1, . . . , u5]⊗ Z[S] representing gener-ators of the corresponding cohomology group. This allows us to determine the ringstructure in H∗(ZS). For example,

[u5u3v4] · [vf ] = [u5u3v4vf ] = 0 = [u5u4v3] · [ve].On the other hand,

[u5u3v4] · [ve] = −[u3u5v4ve] = −[u3u4v5ve]= [u3u4v5vf ] = [u5u4v3vf ] = [u5u4v3] · [vf ].

Here we have used the relations d(u3u4u5ve) = u3u4v5ve − u3u5v4ve andd(u1u3u4v2v5) = u3u4v5ve+u3u4v5vf . All nontrivial products come from Poincareduality. These calculations are summarised by the cohomology ring isomorphism

H∗(ZS) ∼= H∗((S3 × S5)#3 # (S4 × S4)#2

)

where the manifold on the right hand side is the connected sum of three copiesof S3 × S5 and two copies of S4 × S4. We expect that the isomorphism above isinduced by a homeomorphism.

Exercises.

4.10.8. Generalise Proposition 4.3.1 to simplicial posets, i.e. establish a ringisomorphism H∗

((CP∞, pt)S

) ∼= Z[S].4.10.9. Construct a homotopy equivalence

h : (CP∞, pt)S≃−→ ETm ×Tm ZS

by extending the argument of Theorem 4.3.2, and deduce that H∗Tm(ZS) ∼= Z[S].

CHAPTER 5

Toric varieties and manifolds

A toric variety is an algebraic variety on which an algebraic torus (C×)n actswith a dense (Zariski open) orbit. An algebraic torus contains a (compact) torusT n, so toric varieties are toric spaces in our usual sense. Toric varieties are de-scribed by combinatorial-geometric objects, rational fans (see Section 2.1), and thecombinatorics of the fan determines the orbit structure of the torus action.

Toric varieties were introduced in 1970 in the pioneering work of Demazure onCremona group. The geometry of toric varieties, or toric geometry, very quicklybecame one of the most fascinating topics in algebraic geometry and found applica-tions in many other mathematical sciences, sometimes distant from each other. Wehave already mentioned the proof for the necessity part of the g-theorem for sim-plicial polytopes given by Stanley. Other remarkable applications include countinglattice points and volumes of lattice polytopes; relations with Newton polytopesand the number of solutions of a system of algebraic equations (after Khovan-skii and Kushnirenko); discriminants, resultants and hypergeometric functions (af-ter Gelfand, Kapranov and Zelevinsky); reflexive polytopes and mirror symme-try for Calabi–Yau toric hypersurfaces and complete intersections (after Batyrev).Standard references on toric geometry include Danilov’s survey [86] and books byOda [250], Fulton [122] and Ewald [108]. The most recent exhaustive accountby Cox, Little and Schenck [84] covers many new applications, including thosementioned above. Without attempting to give another review of toric geometry,in this chapter we collect the basic definitions and constructions, and emphasisetopological and combinatorial aspects of toric varieties.

We review the three main approaches to toric varieties in the appropriate sec-tions: the ‘classical’ construction via fans, the ‘algebraic quotient’ construction, andthe ‘symplectic reduction’ construction. The intersection of Hermitian quadrics ap-pearing in the symplectic construction of toric varieties links toric geometry tomoment-angle complexes. This link will be developed further in the next chapter.

A basic knowledge of algebraic geometry would much help the reader of thischapter, although it is not absolutely necessary.

5.1. Classical construction from rational fans

An algebraic torus is a commutative complex algebraic group isomorphic to aproduct (C×)n of copies of the multiplicative group C× = C \ 0. It contains acompact torus T n as a Lie (but not algebraic) subgroup.

We shall often identify an algebraic torus with the standard model (C×)n.

Definition 5.1.1. A toric variety is a normal complex algebraic variety Vcontaining an algebraic torus (C×)n as a Zariski open subset in such a way thatthe natural action of (C×)n on itself extends to an action on V .

185

186 5. TORIC VARIETIES AND MANIFOLDS

It follows that (C×)n acts on V with a dense orbit. Toric varieties originally ap-peared as equivariant compactifications of an algebraic torus, although non-compact(e.g., affine) examples are now of equal importance.

Example 5.1.2. The algebraic torus (C×)n and the affine space Cn are thesimplest examples of toric varieties. A compact example is given by the projectivespace CPn on which the torus acts in homogeneous coordinates as follows:

(t1, . . . , tn) · (z0 : z1 : . . . : zn) = (z0 : t1z1 : . . . : tnzn).

Algebraic geometry of toric varieties is translated completely into the languageof combinatorial and convex geometry. Namely, there is a bijective correspondencebetween rational fans in n-dimensional space (see Section 2.1) and complex n-dimensional toric varieties. Under this correspondence,

cones ←→ affine varieties

complete fans ←→ compact (complete) varieties

normal fans of polytopes←→ projective varieties

regular fans ←→ nonsingular varieties

simplicial fans ←→ orbifolds

We review this construction below; the details can be found in the sources men-tioned above. Following the algebraic tradition, we use the coordinate-free notation.

We fix a lattice N of rank n (isomorphic to Zn), and denote by NR its ambientn-dimensional real vector space N ⊗Z R ∼= Rn. Define the algebraic torus C×

N =N ⊗Z C× ∼= (C×)n. All cones and fans in this chapter are rational.

Construction 5.1.3. We first describe how to assign an affine toric varietyto a cone σ ⊂ NR. Consider the dual cone σv ⊂ N∗

R (see (2.1)) and denote by

Sσ = σv ∩N∗

the set of its lattice points. Then Sσ is a finitely generated semigroup (with respectto addition). Let Aσ = C[Sσ] be the semigroup ring of Sσ. It is a commutativefinitely generated C-algebra, with a C-vector space basis χu : u ∈ Sσ. Themultiplication in Aσ is defined via the addition in Sσ:

χu · χu ′

= χu+u ′

,

so χ0 is the unit. The affine toric variety Vσ corresponding to σ is the affinealgebraic variety corresponding to Aσ:

Vσ = Spec(Aσ), Aσ = C[Vσ].

By choosing a multiplicative generator set in Aσ we represent it as a quotient

Aσ = C[x1, . . . , xr ]/I;then the variety Vσ is the common zero set of polynomials from the ideal I. Eachpoint of Vσ corresponds to a semigroup homomorphism Homsg(Sσ,Cm), whereCm = C× ∪ 0 is the multiplicative semigroup of complex numbers.

Now if τ is a face of σ, then σv ⊂ τ v, and the inclusion of semigroup algebrasC[Sσ]→ C[Sτ ] induces a morphism Vτ → Vσ, which is an inclusion of a Zariski opensubset. This allows us to glue the affine varieties Vσ corresponding to all cones σ ina fan Σ into an algebraic variety VΣ, referred to as the toric variety corresponding

5.1. CLASSICAL CONSTRUCTION FROM RATIONAL FANS 187

to the fan Σ. More formally, VΣ may be defined as the colimit of algebraic varietiesVσ over the partially ordered set of cones of Σ:

VΣ = colimσ∈Σ

Vσ.

Here is the crucial point: the fact that the cones σ patch into a fan Σ guaranteesthat the variety VΣ obtained by gluing the pieces Vσ is Hausdorff in the usualtopology. In algebraic geometry, the Hausdorffness is replaced by the related notionof separatedness: a variety V is separated if the image of the diagonal map ∆ : V →V × V is Zariski closed. A separated variety is Hausdorff in the usual topology.

Lemma 5.1.4. If a collection of cones σ forms a fan Σ, then the varietyVΣ = colimσ∈Σ Vσ is separated.

Proof. Using the separatedness criterion of [285, Ch. V, §4.3] (see also [86,Proposition 5.4]), it is enough to verify the following: if cones σ and σ′ intersectin a common face τ , then the diagonal map Vτ → Vσ × Vσ′ is a closed embedding.This is equivalent to the assertion that the natural homomorphism Aσ ⊗Aσ′ → Aτis surjective. To prove this, we use Separation Lemma (Lemma 2.1.2). Accordingto it, there is a linear function u which is nonnegative on σ, nonpositive on σ′ andthe intersection of the hyperplane u⊥ with σ is τ . Now take u ′ ∈ τ v, i.e. u ′ isnonnegative on τ . Then there is an integer k > 0 such that u ′ + ku is nonnegativeon σ, i.e. u ′ + ku ∈ σv. Then u ′ = (u ′ + ku) + (−ku) ∈ σv + σ′v. It followsthat Sσ ⊕ Sσ′ → Sτ is surjective map of vector spaces (or semigroups), henceAσ ⊗Aσ′ → Aτ is a surjective homomorphism.

We shall consider only separated varieties in what follows.The variety Vσ carries an algebraic action of the torus C×

N = N ⊗Z C×

(5.1) C×N × Vσ → Vσ, (t , x) 7→ t · x

which is defined as follows. A point t ∈ C×N is determined by a group homomor-

phism N∗ → C×. In coordinates, the homomorphism Zn ∼= N∗ → C× correspond-ing to t = (t1, . . . , tn) is given by

u = (u1, . . . , un) 7→ t(u) = (tu11 · · · tun

n ).

A point x ∈ Vσ corresponds to a semigroup homomorphism Sσ → Cm. Thenwe define t · x as the point in Vσ corresponding to the semigroup homomorphismSσ → Cm given by

u 7→ t(u)x(u).

The homomorphism of algebras Aσ → Aσ ⊗ C[N∗] dual to the action (5.1) mapsχu to χu ⊗ χu for u ∈ Sσ. If σ = 0, then we obtain the multiplication inthe algebraic group C×

N . The actions on the varieties Vσ are compatible with theinclusions of open sets Vτ → Vσ corresponding to the inclusions of faces τ ⊂ σ.Therefore, for each fan Σ we obtain a C×

N -action on the variety VΣ, which extends

the C×N -action on itself.

Example 5.1.5. Let N = Zn and let σ be the cone spanned by the first k basisvectors e1, . . . , ek, where 0 6 k 6 n. The semigroup Sσ = σv ∩N∗ is generated bythe dual elements e∗

1, . . . , e∗k and ±e∗

k+1, . . . ,±e∗n. Therefore,

Aσ ∼= C[x1, . . . , xk, xk+1, x−1k+1, . . . , xn, x

−1n ],


where we set xi = χe∗i . It follows that the corresponding affine variety is

Vσ ∼= C× · · · × C× C× × · · · × C× = Ck × (C×)n−k.

In particular, for k = n we obtain an n-dimensional affine space, and for k = 0 (i.e.σ = 0) we obtain the algebraic torus (C×)n.

Example 5.1.6. Let σ ⊂ R2 be the cone generated by the vectors e2 and2e1 − e2 (note that these two vectors do not span Z2, so this cone is not regular).The dual cone σv is generated by e∗

1 and e∗1 +2e∗

2. The semigroup Sσ is generatedby e∗

1, e∗1 + e∗

2 and e∗1 + 2e∗

2, with one relation among them. Therefore,

Aσ = C[x, xy, xy2] ∼= C[u, v, w]/(v2 − uw)and Vσ is a quadratic cone (a singular variety).

e1

e2

−e1 − e2

(a)

e1

e2

−e2

−e1 + ke2

(b)

Figure 5.1. Complete fans in R2

Example 5.1.7. Let Σ be the complete fan in R2 with the following threemaximal cones: the cone σ0 generated by e1 and e2, the cone σ1 generated by e2

and −e1 − e2, and the cone σ2 generated by −e1 − e2 and e1, see Fig. 5.1 (a).Then each affine variety Vσi

is isomorphic to C2, with coordinates (x, y) for σ0,(x−1, x−1y) for σ1, and (y−1, xy−1) for σ2. These three affine charts glue togetherinto the complex projective plane VΣ = CP 2 in the standard way: if (z0 : z1 : z2)are the homogeneous coordinats in CP 2, then we have x = z1/z0 and y = z2/z0.

Example 5.1.8. Fix k ∈ Z and consider the complete fan in R2 with thefour two-dimensional cones generated by the pairs of vectors (e1, e2), (e1,−e2),(−e1 + ke2,−e2) and (−e1 + ke2, e2), see Fig 5.1 (b). It can be shown that thecorresponding toric variety Fk is the projectivisation CP (C⊕O(k)) of the sum of atrivial line bundle C and the kth power O(k) = γ⊗k of the canonical line bundle γover CP 1 (an exercise). These 2-dimensional complex varieties Fk are known asHirzebruch surfaces.

Example 5.1.9. Let K be a simplicial complex on [m]. The complement U(K)of the coordinate subspace arrangement corresponding to K (see (4.22)) is a (C×)m-invariant subset in Cm, and therefore it is a nonsingular toric variety. This varietyis not affine in general; it is quasiaffine (the complement to a Zariski closed subsetin an affine variety).

The fan ΣK corresponding to U(K) consists of the cones σI ⊂ Rm generated bythe basis vectors ei1 , . . . , e ik , for all simplices I = i1, . . . , ik ∈ K. The affine toricvariety corresponding to σI is (C,C×)I , and the affine cover of U(K) is its polyhedralproduct decomposition U(K) = ⋃I∈K(C,C

×)I given by Proposition 4.7.3.

5.2. PROJECTIVE TORIC VARIETIES AND POLYTOPES 189

The inclusion poset of closures of C×N -orbits of VΣ is isomorphic to the re-

versed inclusion poset of faces of Σ. That is, k-dimensional cones of Σ corre-spond to codimension-k orbits of the algebraic torus action on VΣ. In particular,n-dimensional cones correspond to fixed points, and the apex (the zero cone) cor-responds to the dense orbit. Furthermore, if a subcollection of cones of Σ forms afan Σ′, then the toric variety VΣ′ is embedded into VΣ as a Zariski open subset.

We recall from Section 2.1 that a fan is called simplicial (respectively, regular)if each of its cones is generated by a part of basis of the space NR (respectively, ofthe lattice N), and a fan is called complete if the union of its cones is the whole NR.

A toric variety VΣ is compact (in usual topology) if and only if the fan Σ iscomplete. If Σ is a simplicial fan, then VΣ is an orbifold , that is, it is locallyisomorphic to a quotient of Cn by a finite group action. A toric variety VΣ isnonsingular (smooth) if and only if the fan Σ is regular.

Exercises.

5.1.10. Let Σ be the ‘multifan’ in R1 consisting of two identical 1-dimensionalcones generated by e1 and a 0-dimensional cone 0. Describe the algebraic varietyVΣ obtained by gluing the affine varieties corresponding to this ‘multifan’ and showthat VΣ is not separated (or non-Hausdorff in the usual topology).

5.1.11. Describe the toric variety corresponding to the fan with 3 one-dimensional cones generated by the vectors e1, e2 and −e1 − e2.

5.1.12. Show that the toric variety of Example 5.1.8 is isomorphic to the Hirze-bruch surface Fk = CP (C⊕O(k)).

5.1.13. Show that the Hirzebruch surface Fk is homeomorphic to S2 × S2 foreven k and is homeomorphic to CP 2#CP 2 for odd k, where # denotes the connectedsum, and CP 2 is CP 2 with the orientation reversed.

5.2. Projective toric varieties and polytopes

Construction 5.2.1 (projective toric varieties). Let P be a convex polytopewith vertices in the dual lattice N∗ (a lattice polytope), and let ΣP be the normal fanof P (see Construction 2.1.3). Since P ⊂ N∗

R, the fan ΣP belongs to the space NR.It has a maximal cone σv for each vertex v ∈ P . The dual cone σ∗

v is the ‘vertexcone’ at v, generated by all vectors pointing from v to other points of P .

Define the toric variety VP = VΣP. Since the normal fan ΣP does not depend on

the linear size of the polytope, we may assume that for each vertex v the semigroupSσv

is generated by the lattice points of the polytope (this can always be achievedby replacing P by kP with sufficiently large k). Since N∗ is the lattice of charactersof the algebraic torus C×

N = N ⊗Z C× (that is, N∗ = HomZ(C×N ,C

×)), the latticepoints of the polytope P ⊂ N∗ define an embedding

iP : C×N → (C×)|N

∗∩P |,

where |N∗ ∩ P | is the number of lattice points in P .

Proposition 5.2.2 (see [84, §2.2] or [122, §3.4]). The toric variety VP is

identified with the projective closure iP (C×N ) ⊂ CP |N∗∩P |.

It follows that toric varieties arising from polytopes are projective, i.e. can bedefined by a set of homogeneous equations in a projective space. The converse is


also true: the fan corresponding to a projective toric variety is the normal fan of alattice polytope.

The polytope P carries more geometric information than the normal fan ΣP :different lattice polytopes with the same normal fan Σ correspond to different pro-jective embeddings of the toric variety VΣ.

Nonsingular projective toric varieties correspond to lattice polytopes P whichare simple and Delzant (that is, for each vertex v, the normal vectors of facetsmeeting at v form a basis of the lattice N).

Constructions of Chapter 1 provide explicit series of Delzant polytopes andtherefore nonsingular projective toric varieties. Basic examples include simplicesand cubes in all dimensions. The product of two Delzant polytopes is obviouslyDelzant. If P is a Delzant polytope, then its face truncation P ∩ H> (Construc-tion 1.1.12) by an appropriately chosen hyperplane H is also Delzant (an exercise).All nestohedra (in particular, permutahedra and associahedra) admit Delzant real-isations. This fact, first observed in [265, Proposition 7.10], can be proved eitherby using the sequence of face truncations described in Lemma 1.5.17 or directlyfrom the presentation of nestohedra given in Proposition 1.5.11.

Example 5.2.3. The fan Σ described in Example 2.1.4 is regular, but cannot beobtained as the normal fan of a simple polytope. The corresponding 3-dimensionaltoric variety VΣ is compact and nonsingular, but not projective.

We note that although the fan Σ from the previous example cannot be realisedgeometrically as the fan over the faces of a simplicial polytope (or, equivalently, asthe normal fan of a simple polytope), its underlying simplicial complex KΣ is never-theless combinatorially equivalent to the boundary complex of a simplicial polytope(namely, an octahedron with a pyramid over one of its facets, see Fig. 2.1). In otherwords, using the terminology of Section 2.5, the starshaped sphere triangulation KΣ

is polytopal (in the combinatorial sense).There are, of course, nonpolytopal starshaped spheres, such as the Barnette

sphere K (see Example 2.5.8). It is easy to see that a simplicial fan realising theBarnette sphere can be chosen rational. However, to realise a nonpolytopal sphereby a regular fan turned out to be a more difficult task. This question has beenfinally settled by Suyama [299]; his example is obtained by subdividing a simplicialfan realising the Barnette sphere. This is also important for the study of qua-sitoric manifolds and other topological generalisations of toric varieties discussedin Chapter 7.

We observe that each combinatorial simple polytope admits a convex realisationas a lattice polytope. Indeed by a small perturbation of the defining inequalitiesin (1.1) we can make all of them rational (that is, with rational a i and bi). Such aperturbation does not change the combinatorial type, as the half-spaces defined bythe inequalities are in general position. As a result, we obtain a simple polytope P ′

of the same combinatorial type with rational vertex coordinates. To get a latticepolytope (say, with vertices in the standard lattice Zn) we just take the magnifiedpolytope kP ′ for appropriate k ∈ Z. Similarly, by perturbing the vertices insteadof the hyperplanes, we can obtain a lattice realisation for an arbitrary simplicialpolytope (and we can obtain a rational fan realisation of any starshaped spheretriangulation). However, this argument does not work for convex polytopes whichare neither simple nor simplicial. In fact, there are exist nonrational combinatorial

5.3. COHOMOLOGY OF TORIC MANIFOLDS 191

polytopes, which cannot be realised with rational vertex coordinates, see [325,Example 6.21] and the discussion there.

Toric geometry, even in its topological part, does not translate to a purelycombinatorial study of fans and polytopes: the underlying convex geometry iswhat really matters. This is illustrated by the simple observation that differentrealisations of a combinatorial polytope by lattice polytopes often produce different(even topologically) toric varieties:

Example 5.2.4. The complete regular fan Σk corresponding to the Hirzebruchsurface Fk (see Fig. 5.1 (b)) is the normal fan of a lattice quadrilateral (trapezoid),e.g. given by

Pk = (x1, x2) ∈ R2 : x1 > 0, x2 > 0, −x1 + kx2 > −1, −x2 > −1.The polytopes Pk corresponding to different k are combinatorially equivalent (asthey are all quadrilaterals), but the topology of the corresponding toric varietiesFk is different for even and odd k (see Exercise 5.1.13).

Furthermore, there exist combinatorial simple polytopes that do not admit anylattice realisation P with smooth VP :

Example 5.2.5 ([90, 1.22]). Let P be the dual of a 2-neighbourly simplicialn-polytope (e.g., a cyclic polytope of dimension n > 4, see Example 1.1.17) withm > 2n vertices. Then for any lattice realisation of P the corresponding normalfan ΣP is not regular, and the toric variety VP is singular. Indeed, assume that thenormal fan ΣP is regular. Since the 1-skeleton of KP is a complete graph, each pairof primitive generators a i,aj of one-dimensional cones of ΣP is a part of basis ofZn, and therefore a i and aj must be different modulo 2. This is a contradiction,since the number of different nonzero vectors in Zn2 is 2n − 1.

Exercises.

5.2.6. Let P be a Delzant polytope and G ⊂ P a face. Show that a hyperplaneH truncating G from P in Construction 1.1.12 can be chosen so that the truncatedpolytope P ∩H> is Delzant.

5.2.7. The presentation of a nestohedron PB from Proposition 1.5.11 is Delzant.

5.2.8. Describe explicitly a rational fan realising the Barnette sphere by writingdown its primitive integral generator vectors.

5.2.9. Write down a system of homogeneous equations defining each Hirzebruchsurface in a projective space. (Hint: use Construction 5.2.1.)

5.3. Cohomology of toric manifolds

A toric manifold is a smooth compact toric variety. (Compactness will be al-ways understood in the sense of usual topology; it corresponds to algebraic geome-ter’s notion of completeness.) Toric manifolds VΣ correspond to complete regularfans Σ. Projective toric manifolds VP correspond to lattice polytopes P whosenormal fans are regular.

The cohomology of a toric manifold VΣ can be calculated effectively from thefan Σ. The Betti numbers are determined by the combinatorics of Σ only, while thering structure of H∗(VΣ) depends on the geometric data. The required combinato-rial ingredients are the h-vector h(KΣ) = (h0, h1, . . . , hn) (see Definition 2.2.5) of


the underlying simplicial complex KΣ and its face ring Z[KΣ] (Definition 3.1.1). Thegeometric data consists of the primitive generators a1, . . . ,am of one-dimensionalcones (edges) of Σ.

Theorem 5.3.1 (Danilov–Jurkiewicz). Let VΣ be the toric manifold correspond-ing to a complete regular fan Σ in NR. The cohomology ring of VΣ is given by

H∗(VΣ) ∼= Z[v1, . . . , vm]/I,where v1, . . . , vm ∈ H2(VΣ) are the cohomology classes dual the invariant divisorscorresponding to the one-dimensional cones of Σ, and I is the ideal generated byelements of the following two types:

(a) vi1 · · · vik with i1, . . . , ik /∈ KΣ (the Stanley–Reisner relations);

(b)

m∑

i=1

〈aj ,u〉vi, for any u ∈ N∗.

The homology groups of VΣ vanish in odd dimensions, and are free abelian ineven dimensions, with ranks given by

b2i(VΣ) = hi(KΣ),

where hi(KΣ), i = 0, 1, . . . , n, are the components of the h-vector of KΣ.

This theorem was proved by Jurkiewicz for projective toric manifolds and byDanilov [86, Theorem 10.8] in the general case. We shall give a topological proofof a more general result in Section 7.4 (see Theorem 7.4.35).

To obtain an explicit presentation of the ring H∗(VΣ) we choose a basis of Nand write the vectors aj in coordinates: aj = (aj1, . . . , ajn)

t, 1 6 j 6 m. Then theideal JΣ is generated by the n linear forms

ti = a1iv1 + · · ·+ amivm ∈ Z[v1, . . . , vm], 1 6 i 6 n.

By Lemma 3.3.2, the sequence t1, . . . , tn is an lsop in the Cohen–Macaulayring Z[KΣ], so it is a regular sequence. Hence, Z[KΣ] is a free Z[t1, . . . , tn]-module,and statement (a) of Theorem 5.3.1 follows from (b) and Theorem 3.1.10.

Remark. Theorem 5.3.1 remains valid for complete simplicial fans and corre-sponding toric orbifolds if the integer coefficients are replaced by the rationals [86].The integral cohomology of toric orbifolds often has torsion, and the ring structureis subtle even in the simplest case of weighted projective spaces [181], [17].

It follows from Theorem 5.3.1 that the cohomology ring of VΣ is generatedby two-dimensional classes. This is the first property to check if one wishes todetermine whether a given algebraic variety or smooth manifold has a structureof a toric manifold. For instance, this rules out flag varieties and Grassmaniansdifferent from projective spaces. Another important property of toric manifolds isthat the Chow ring of VΣ coincides with its integer cohomology ring [122, § 5.1].

Assume now that Σ = ΣP is the normal fan of a lattice polytope P = P (A, b)given by (1.1). Let VP be the corresponding projective toric variety, see Con-struction 5.2.1. In the notation of Theorem 5.3.1, the linear combination DP =b1D1 + · · ·+ bmDm is an ample divisor on VP (see, e.g., [84, Proposition 6.1.10]).This means that, when k is sufficiently large, kDP is a hyperplane section divisorfor a projective embedding VP ⊂ CP r. In fact, the space of sections H0(VP , kDP )of (the line bundle corresponding to) kDP has basis corresponding to the latticepoints in kP . One may take k so that kP has ‘enough lattice points’ to get an

5.3. COHOMOLOGY OF TORIC MANIFOLDS 193

embedding of VP into the projectivisation of H0(VP , kDP ); this is exactly the em-bedding described in Construction 5.2.1. Let ω = b1v1 + · · · + bmvm ∈ H2(VP ;C)be the complex cohomology class of DP .

Theorem 5.3.2 (Hard Lefschetz Theorem for toric orbifolds). Let P be a latticesimple polytope (1.1), let VP be the corresponding projective toric variety, and letω = b1v1 + · · ·+ bmvm ∈ H2(VP ;C) be the class defined above. Then the maps

Hn−i(VP ;C)ωi

−→ Hn+i(VP ;C)

are isomorphisms for all i = 1, . . . , n.

If VP is smooth, then it is Kahler, and ω is the class of the Kahler 2-form.The proof of the Hard Lefschetz Theorem is well beyond the scope of this book.

In fact, it is a corollary of a more general version of Hard Lefschetz Theorem forthe (middle perversity) intersection cohomology, which is valid for all projectivevarieties (not necessary orbifolds). See the discussion in [122, §5.2] or [84, §12.6].

Now we are ready to give Stanley’s argument for the ‘only if’ part of the g-theorem for simple polytopes:

Proof of the necessity part of Theorem 1.4.14. We need to establishconditions (a)–(c) for a combinatorial simple polytope. Realise it by a latticepolytope P ⊂ Rn as described in the previous section. Let VP be the corre-sponding toric variety. Part (a) is already proved (Theorem 1.3.4). It followsfrom Theorem 5.3.2 that the multiplication by ω ∈ H2(VP ;Q) is a monomorphismH2i−2(VP ;Q) → H2i(VP ;Q) for i 6

[n2

]. This together with part (a) of Theo-

rem 5.3.1 gives that hi−1 6 hi for 0 6 i 6[n2

], thus proving (b). To prove (c),

define the graded commutative Q-algebra A = H∗(VP ;Q)/(ω), where (ω) is theideal generated by ω. Then A0 = Q, A2i = H2i(VP ;Q)/(ω · H2i−2(VP ;Q)) for1 6 i 6

[n2

], and A is generated by degree-two elements (since so is H∗(VP ;Q)). It

follows from Theorem 1.4.12 that the numbers dimA2i = hi − hi−1, 0 6 i 6[n2

],

are the components of an M -vector, thus proving (c) and the whole theorem.

Remark. The Dehn–Sommerville equations now can be interpreted asPoincare duality for VP . (Even though VP may be not smooth, the rational coho-mology algebra of a toric orbifold satisfies Poincare duality.)

The Hard Lefschetz Theorem holds for projective varieties only. Therefore,Stanley’s argument cannot be generalised to nonpolytopal spheres. However, thecohomology of toric varieties can be used to prove statements generalising the g-theorem in a different direction, namely, to the case of general (not necessarilysimple or simplicial) convex polytopes. So suppose P is a convex lattice n-polytope,and VP is the corresponding projective toric variety. If P is not simple then VP hasworse than orbifold-type singularities and its ordinary cohomology behaves badly.The Betti numbers of VP are not determined by the combinatorial type of P and do

not satisfy Poincare duality. On the other hand, the dimensions hi = dim IH2i(VP )of the intersection homology groups of VP are combinatorial invariants of P (seethe description in [291] or [84, §12.5]). The vector

h(P ) = (h0, h1, . . . , hn)

is called the intersection h-vector, or the toric h-vector of P . If P is simple, then

the toric h-vector coincides with the standard h-vector, but in general h(P ) is


not determined by the face numbers of P . The toric h-vector satisfies the ‘Dehn–

Sommerville equations’ hi = hn−i, and the Hard Lefschetz Theorem for intersectioncohomology shows that it also satisfies the GLBC inequalities:

h0 6 h1 6 · · · 6 h[n2

].

In the case when P cannot be realised by a lattice polytope (i.e., when P is non-rational), the toric h-vector can still be defined combinatorially, but the GLBCinequalities require a separate proof. Partial results in this direction were obtainedby several people, before the Hard Lefschetz Theorem for nonrational polytopeswas eventually proved in the work of Karu [179]. This result also gives a purelycombinatorial proof of the Hard Lefschetz Theorem for projective toric varieties.

Exercises.

5.3.3. Show that the complex Grassmanian Grk(Cn) (k-planes in Cn) with1 < k < n − 1 does not support an algebraic torus action turning it into a toricvariety.

5.4. Algebraic quotient construction

Along with the classical construction of toric varieties from fans, described inSection 5.1, there is an alternative way to define a toric variety: as the quotientof a Zariski open subset in Cm (more precisely, the complement of a coordinatesubspace arrangement) by an action of an abelian algebraic group (a product of analgebraic torus and a finite group). Different versions of this construction, whichwe refer to as simply the ‘quotient construction’, have appeared in the work ofseveral authors since the early 1990s. In our exposition we mainly follow the workof Cox [83] (and also its modernised exposition in [84, Chapter 5]); more historicalremarks can be also found in these sources.

Quotients in algebraic geometry. Taking quotients of algebraic varieties byalgebraic group actions is tricky for both topological and algebraic reasons. First,as algebraic groups are non-compact (as algebraic tori), their orbits may be notclosed, and the quotients may be non-Hausdorff. Second, even if the quotient isHausdorff as a topological space, it may fail to be an algebraic variety. This maybe remedied to some extent by the notion of the categorical quotient.

Let X be an algebraic variety with an action of an affine algebraic group G.An algebraic variety Y is called a categorical quotient of X by the action of G ifthere exists a morphism π : X → Y which is constant on G-orbits of X and hasthe following universal property: for any morphism ϕ : X → Z which is constanton G-orbits, there is a unique morphism ϕ : Y → Z such that ϕ π = ϕ. This isdescribed by the diagram

Xϕ //

π

Z

Y

ϕ??⑦

⑦⑦

⑦

A categorical quotient Y is unique up to isomorphism, and we denote it by X//G.Assume first that X = SpecA is an affine variety, where A = C[X ] is the

algebra of regular functions on X . Let C[X ]G be the subalgebra of G-invariantfunctions (i.e. such functions f that f(gx) = f(x) for any g ∈ G and x ∈ X).

5.4. ALGEBRAIC QUOTIENT CONSTRUCTION 195

If G is an algebraic torus (or any reductive affine algebraic group), then C[X ]G isfinitely generated. The corresponding affine variety SpecC[X ]G is the categoricalquotient X//G. The quotient morphism π : X → X//G is dual to the inclusionof algebras C[X ]G → C[X ]. The morphism π is surjective and induces a one-to-one correspondence between points of X//G and closed G-orbits of X (i.e. π−1(x)contains a unique closed G-orbit for any x ∈ X//G, see [84, Proposition 5.0.7]).

Therefore, if all G-orbits of an affine variety X are closed, then the categoricalquotient X//G is identified as a topological space with the ordinary ‘topological’quotientX/G. Quotients of this type are called geometric and also denoted byX/G.

Example 5.4.1. Let C× act on C = Spec(C[z]) by scalar multiplication. Thereare two orbits: the closed orbit 0 and the open orbit C×. The topological quotientC/C× is a non-Hausdorff two-point space.

On the other hand, the categorical quotient C//C× = Spec(C[z]C×

) is a point,since any C×-invariant polynomial is constant (and there is only one closed orbit).

Similarly, if C× acts on Cn = Spec(C[z1, . . . , zn]) diagonally, then an invari-ant polynomial satisfies f(λz1, . . . , λzn) = f(z1, . . . , zn) for all λ ∈ C×. Such apolynomial must be constant, hence Cn//C× is a point.

In good cases categorical quotients of general (non-affine) varieties X may beconstructed by ‘gluing from affine pieces’ as follows. Assume that G acts on X andπ : X → Y is a morphism of varieties that is constant on G-orbits. If Y has anopen affine cover Y =

⋃α Vα such that π−1(Vα) is affine and Vα is the categorical

quotient (that is, π|π−1(Vα) : π−1(Vα) → Vα is the morphism dual to the inclusion

of algebras C[π−1(Vα)]G → C[π−1(Vα)]), then Y is the categorical quotient X//G.

Example 5.4.2. Let C× act on C2\0 diagonally, where C2 = Spec(C[z0, z1]).We have an open affine cover C2 \ 0 = U0 ∪ U1, where

U0 = C2 \ z0 = 0 = C× × C = Spec(C[z±10 , z1]),

U1 = C2 \ z1 = 0 = C× C× = Spec(C[z0, z±11 ]),

U0 ∩ U1 = C2 \ z0z1 = 0 = C× × C× = Spec(C[z±10 , z±1

1 ]).

The algebras of C×-invariant functions are

C[z±10 , z1]

C×

= C[z1/z0], C[z0, z±11 ]C

×

= C[z0/z1], C[z±10 , z±1

1 ]C×

= C[(z1/z0)±1].

It follows that V0 = U0//C× ∼= C and V1 = U1//C× ∼= C glue together alongV0 ∩V1 = (U0 ∩U1)//C× ∼= C× in the standard way to produce CP 1. All C×-orbitsare closed in C2 \ 0, hence CP 1 = (C2 \ 0)/C× is the geometric quotient.

Similarly, CPn = (Cn+1 \ 0)/C× for the diagonal action of C×.

Example 5.4.3. Now we let C× act on C2 \ 0 by λ · (z0, z1) = (λz0, λ−1z1).

Using the same affine cover of C2 \ 0 as in the previous example, we obtain thefollowing algebras of C×-invariant functions:

C[z±10 , z1]

C×

= C[z0z1], C[z0, z±11 ]C

×

= C[z0z1], C[z±10 , z±1

1 ]C×

= C[(z0z1)±1].

This times gluing together V0 and V1 along V0∩V1 ∼= C× gives the variety obtainedfrom two copies of C by identifying all nonzero points. This variety is nonseparated(the two zeros do not have nonintersecting neighbourhoods in the usual topology).Since we only consider separated varieties, this is not a categorical quotient.


A toric variety VΣ will be described as the quotient of the ‘total space’ U(Σ)by an action of a commutative algebraic group G. We now proceed to describe Gand U(Σ).

The total space U(Σ) and the acting group G. Let Σ be a rational fan inthe n-dimensional space NR with m one-dimensional cones generated by primitivevectors a1, . . . ,am. We shall assume that the linear span of a1, . . . ,am is thewhole NR. (Equivalently, the toric variety VΣ does not have torus factors, i.e.cannot be written as VΣ = VΣ′ × C×. For the general case see [84, §5.1].)

We consider the map of lattices A : Zm → N sending the ith basis vector of Zm

to a i ∈ N . Our assumption implies that the corresponding map of algebraic tori,

A⊗Z C× : (C×)m → C×N

is surjective.Define the group G = G(Σ) as the kernel of the map A⊗ZC×, which we denote

by expA. We therefore have an exact sequence of groups

(5.2) 1 −→ G −→ (C×)mexpA−→ C×

N −→ 1.

Explicitly, G is given by

(5.3) G =(z1, . . . , zm) ∈ (C×)m :

m∏

i=1

z〈ai,u〉i = 1 for any u ∈ N∗

.

The group G is isomorphic to a product of (C×)m−n and a finite abelian group. IfΣ is a regular fan with at least one n-dimensional cone, then G ∼= (C×)m−n.

Given a cone σ ∈ Σ, we set g(σ) = i1, . . . , ik ⊂ [m] if σ is generated byai1 . . . ,a ik , and consider the monomial zσ =

∏j /∈g(σ) zj. The quasiaffine variety

U(Σ) = Cm \ z ∈ Cm : zσ = 0 for all σ ∈ Σhas the affine cover

(5.4) U(Σ) =⋃

σ∈Σ

U(σ).

by affine varieties

U(σ) = z ∈ Cm : zσ 6= 0 = z ∈ Cm : zj 6= 0 for j /∈ g(σ) = (C,C×)g(σ).

Here we used the notation of Construction 4.2.1, so that U(σ) ∼= Ck × (C×)m−k.Each subset U(σ) ⊂ Cm is invariant under the coordinatewise action of (C×)m onCm, so that U(Σ) is also invariant.

By definition, U(Σ) is the complement of a union of coordinate subspaces, sowe know from Proposition 4.7.2 that it has the form

(5.5) U(K) = Cm∖ ⋃

i1,...,ik/∈K

z ∈ Cm : zi1 = · · · = zik = 0

= ZK(C,C

×)

for some simplicial complex K on [m]. What is this simplicial complex?The answer is suggested by decomposition (5.4). We define the simplicial com-

plex KΣ generated by all subsets g(σ) ⊂ [m]:

KΣ = I : I ⊂ g(σ) for some σ ∈ Σ.If Σ is a simplicial fan, then each I ⊂ g(σ) is g(τ) for some τ ∈ Σ, and we obtain the‘underlying complex’ of Σ defined in Example 2.2.7. In particular, if Σ is simplicial


and complete, then KΣ is a triangulation of Sn−1; and if Σ is a normal fan of asimple polytope, then KΣ is the boundary complex of the dual simplicial polytope.If Σ is the normal fan of a non-simple polytope P (i.e. the fan over the faces of thepolar polytope P ∗), then KΣ is obtained by replacing each face of P ∗ by a simplexwith the same set of vertices; such a simplicial complex is not pure in general.

Proposition 5.4.4. We have U(Σ) = U(KΣ).

Proof. We have (C,C×)I ⊂ (C,C×)g(σ) whenever I ⊂ g(σ), hence,

U(Σ) =⋃

σ∈Σ

U(σ) =⋃

σ∈Σ

(C,C×)g(σ) =⋃

σ∈Σ,I⊂g(σ)

(C,C×)I =⋃

I∈KΣ

(C,C×)I = U(KΣ).

We observe that the subset U(Σ) ⊂ Cm depends only on the combinatorialstructure of the fan Σ, while the subgroup G ⊂ (C×)m depends on the geometricdata, namely, the primitive generators of one-dimensional cones.

Toric variety as a quotient. Since U(Σ) ⊂ Cm is invariant under the coor-dinatewise action of (C×)m, we obtain a G-action on U(Σ) by restriction.

Theorem 5.4.5 (Cox [83, Theorem 2.1]). Assume that the linear span of one-dimensional cones of Σ is the whole space NR.(a) The toric variety VΣ is naturally isomorphic to the categorical quotient U(Σ)//G.(b) VΣ is the geometric quotient U(Σ)/G if and only if the fan Σ is simplicial.

Proof. We first prove that the affine variety Vσ corresponding to a cone σ ∈Σ is the categorical quotient U(σ)//G. The algebra of regular functions C[U(σ)]is isomorphic to C[zi, z

−1j : 1 6 i 6 m, j /∈ g(σ)] and is generated by Laurent

monomials∏mi=1 z

kii with ki > 0 for i ∈ g(σ).

It follows easily from (5.3) that a monomial∏mi=1 z

kii is invariant under the

G-action on U(σ) if and only if it has the form∏mi=1 z

〈u,a i〉i for some u ∈ N∗.

Conditions 〈u ,a i〉 > 0 for i ∈ g(σ) specify the dual cone σv ⊂ N∗R, see (2.1).

Hence the invariant subalgebra C[U(σ)]G is isomorphic to C[σv∩N∗] = Aσ = C[Vσ]

(the isomorphism is given by∏mi=1 y

〈u,ai〉i 7→ χu ). Thus, U(σ)//G ∼= Vσ.

The next step is to glue the isomorphisms U(σ)//G ∼= Vσ together into anisomorphism U(Σ)//G ∼= VΣ. To do this we need to check that the isomorphismsC[U(σ)]G → C[Vσ] are compatible when we pass to the faces of σ. In other words,for each face τ ⊂ σ we need to establish the commutativity of the diagram

(5.6)

C[U(σ)]G −−−−→ C[U(τ)]G

∼=

yy∼=

C[Vσ ] −−−−→ C[Vτ ].

By the definition of a face, we have τ = σ ∩ u⊥ for some u ∈ σv ∩ N∗, whereu⊥ denotes the hyperplane in NR normal to u . Consider the monomial z (u) =∏mi=1 z

〈u,ai〉i . Since τ = σ ∩ u⊥, the monomial z (u) has positive exponent of

zi for i ∈ g(σ) \ g(τ) and zero exponent of zj for j ∈ g(τ). It follows that thealgebra C[U(τ)] is the localisation of C[U(σ)] by the ideal generated by z (u), i.e.C[U(τ)] = C[U(σ)]z (u). Since z (u) is a G-invariant monomial, the localisation


commutes with passing to invariant subalgebras, i.e. C[U(τ)]G = C[U(σ)]Gz (u).

Similarly, C[Vτ ] = Aτ = (Aσ)χu . Diagram (5.6) then takes the form

C[U(σ)]G −−−−→ C[U(σ)]Gz (u)

∼=

yy∼=

C[Vσ ] −−−−→ C[Vσ]χu ,

where the vertical arrows are localisation maps. It is obviously commutative.Now using the affine cover (5.4) and the compatibility of the isomorphisms

on affine varieties we obtain the isomorphism U(Σ)//G ∼= VΣ =⋃σ∈Σ Vσ. State-

ment (a) is therefore proved.

To verify (b) we need to check that all orbits of the G-action on U(Σ) are closedif and only if the fan Σ is simplicial.

Assume Σ is simplicial, and consider any G-orbit Gz , z ∈ U(Σ). We shallprove that Gz is closed in the usual topology, which is sufficient since the closures oforbits in the usual and Zariski topologies coincide. We need to check that whenevera sequence w (k) : k = 1, 2, . . . of points of Gz has a limit w ∈ U(Σ), this limitis in Gz . Write w (k) = g (k)z with g (k) ∈ G. Then it is enough to show that asubsequence of g (k) converges to a point g ∈ G, as in this case limk→∞ w (k) =gz ∈ Gz . We write

g (k) =(g(k)1 , . . . , g(k)m

)=(eα

(k)1 +iβ

(k)1 , . . . , eα

(k)m +iβ(k)

m

)∈ G ⊂ (C×)m

where g(k)j ∈ C× and α

(k)j , β

(k)j ∈ R. Since eiβ

(k)j ∈ S and the circle is compact, we

may assume by passing to a subsequence that the sequence eiβ(k)j has a limit eiβj

as k→∞ for each j = 1, . . . ,m. It remains to consider the sequences eα(k)j .

By passing to a subsequence we may assume that each sequence α(k)j , j =

1, . . . ,m, has a finite or infinite limit (including ±∞). Let

I+ = j : α(k)j → +∞ ⊂ [m], I− = j : α(k)

j → −∞ ⊂ [m].

Since the sequence w (k) = g (k)z is converging to w = (w1, . . . , wm) ∈ U(Σ),we have zj = 0 for j ∈ I+ and wj = 0 for j ∈ I−. Then it follows from thedecomposition U(Σ) =

⋃I∈KΣ

(C,C×)I that I+ and I− are simplices of KΣ. Let

σ+, σ− be the corresponding cones of Σ (here we use the fact that Σ is simplicial).Then σ+ ∩ σ− = 0 by definition of a fan. By Lemma 2.1.2, there is a linearfunction u ∈ N∗ such that 〈u ,a〉 > 0 for any nonzero a ∈ σ+, and 〈u ,a〉 < 0 forany nonzero a ∈ σ−. Now, since g (k) ∈ G, it follows from (5.3) that

(5.7)m∑

j=1

α(k)j 〈u ,aj〉 = 0.

This implies that both I+ and I− are empty, as otherwise the sum above tends

to infinity. Thus, each sequence α(k)j has a finite limit αj , and a subsequence of

g (k) converges to (eα1+iβ1 , . . . , eαm+iβm). Passing to the limit in (5.7) and in the

similar equation for β(k)j as k →∞ we obtain that (eα1+iβ1 , . . . , eαm+iβm) ∈ G.

The fact that the G-action on U(Σ) with non-simplicial Σ has non-closed orbitsis left as an exercise (alternatively, see [83, §2] or [84, Theorem 5.1.11].

The quotient torus C×N = (C×)m/G acts on VΣ = U(Σ)//G with a dense orbit.


Remark. Observe that U(Σ) is itself a toric variety by Example 5.1.9. Themap A : Rm → NR, e i 7→ a i, projects the fan ΣK corresponding to U(Σ) to thefan Σ corresponding to VΣ. This projection defines a morphism of toric varietiesU(Σ) → VΣ, which is exactly the quotient described above. Both fans ΣK and Σhave the same underlying simplicial complex K.

Another way to see that the orbits of the G-action on U(Σ) are closed is to usethe almost freeness of this action (see Exercise 5.4.14):

Proposition 5.4.6.

(a) If Σ is a simplicial fan, then the G-action on U(Σ) is almost free (i.e., allstabiliser subgroups are finite);

(b) If Σ is regular, then the G-action on U(Σ) is free.

Proof. The stabiliser of a point z ∈ Cm under the action of (C×)m is

(C×)ω(z ) = (t1, . . . , tm) ∈ (C×)m : ti = 1 if zi 6= 0,where ω(z ) be the set of zero coordinates of z . The stabiliser of z under the G-action is Gz = (C×)ω(z )∩G. Since G is the kernel of the map expA : (C×)m → C×

N

induced by the map of lattices A : Zm → N , the subgroup Gz is the kernel of thecomposite map

(5.8) (C×)ω(z ) → (C×)mexpA−→ C×

N .

This homomorphism of tori is induced by the map of lattices Zω(z ) → Zm → N ,where Zω(z ) → Zm is the inclusion of a coordinate sublattice.

Now let Σ be a simplicial fan and z ∈ U(Σ). Then ω(z ) = g(σ) for a cone σ ∈ Σ.Therefore, the set of primitive generators a i : i ∈ ω(z ) is linearly independent.Hence, the map Zω(z ) → Zm → N taking e i to a i is a monomorphism, whichimplies that the kernel of (5.8) is a finite group.

If the fan Σ is regular, then ai : i ∈ ω(z ) is a part of basis of N . In thiscase (5.8) is a monomorphism and Gz = 1.

Remark. The closedness of orbits is a necessary condition for the topologicalquotient U(Σ)/G to be Hausdorff (Exercise B.3.4). The proof of Theorem 5.4.5 (b)above uses Separation Lemma (Lemma 2.1.2); this is another example of situationwhen convex-geometric separation translates into Hausdorffness.

We may consider the following more general setup. Let a1, . . . ,am be a set ofprimitive vectors in N , and let K be a simplicial complex on [m]. Assume furtherthat for any I ∈ K the set of vectors ai : i ∈ I is linearly independent. The latterset spans a simplicial cone, which we denote by σI . The data K,a1, . . . ,amdefines the coordinate subspace arrangement complement U(K) ⊂ Cm and thegroup G (5.3). Furthermore, the action of G on U(K) is almost free (and it is freeif all cones σI are regular; this is proved in the same way as Proposition 5.4.6).However, the quotient U(K)/G is Hausdorff precisely when the cones σI : I ∈ Kform a fan Σ (see [6, Proposition II.3.1.6] and use the previous remark). In thiscase K = KΣ and U(K) = U(Σ).

To see that the quotient U(Σ)/G is Hausdorff in the case when Σ is a simplicialfan, it is not necessary to use the algebraic criterion of separatedness as in theproof of Lemma 5.1.4. Instead, we may modify the argument for the closedness oforbits in the proof of Theorem 5.4.5 and show that action of G on U(Σ) is proper(Exercise 5.4.15), which guarantees that U(Σ)/G is Hausdorff (Exercise 5.4.16).


Example 5.4.7. Let Vσ be the affine toric variety corresponding to an n-dimensional simplicial cone σ. We may write Vσ = VΣ where Σ is the simplicialfan consisting of all faces of σ. Then m = n, U(Σ) = Cn, and A : Zn → N is themonomorphism onto the full rank sublattice generated by a1, . . . ,an. Therefore,G is a finite group and Vσ = Cn/G = SpecC[z1, . . . , zn]

G.In particular, if we consider the cone σ generated by 2e1−e2 and e2 in R2 (see

Example 5.1.6), then G is Z2 embedded as (1, 1), (−1,−1) in (C×)2. The quotientconstruction realises the quadratic cone Vσ = SpecC[z1, z2]G = SpecC[z21 , z1z2, z

22 ]

as a quotient of C2 by Z2.

Example 5.4.8. Consider the complete fan of Example 5.1.7. Then

U(Σ) = C3 \ z1 = z2 = z3 = 0 = C3 \ 0The subgroup G defined by (5.2) is the diagonal C× in (C×)3. We therefore obtainVΣ = U(Σ)/G = CP 2.

Example 5.4.9. Consider the fan Σ in R2 with three one-dimensional conesgenerated by the vectors e1, e2 and −e1 − e2. This fan is not complete, but itsone-dimensional cones generate R2, so we may apply Theorem 5.4.5. The simplicialcomplex KΣ consists of 3 disjoint points. The space U(Σ) = U(KΣ) is therefore thecomplement to 3 coordinate lines in C3:

U(Σ) = C3∖(z1 = z2 = 0 ∪ z1 = z3 = 0 ∪ z2 = z3 = 0

)

The group G is the diagonal C× in (C×)3. Hence VΣ = U(Σ)/G is a quasiprojectivevariety obtained by removing three points from CP 2.

Exercises.

5.4.10. The one-dimensional cones of a fan Σ ⊂ NR span NR if and only if thetoric variety VΣ does not have C×-factors.

5.4.11. If Σ is a regular fan of full dimension, then G ∼= (C×)m−n.

5.4.12. A G-invariant monomial has the form∏mi=1 z

〈u,ai〉i for some u ∈ N∗.

5.4.13. If the fan Σ is non-simplicial, then there exists a non-closed orbit of theG-action on U(Σ).

5.4.14. Assume that the action of an algebraic group G on an algebraic varietyX is almost free. Show that then all G-orbits are Zariski closed.

5.4.15. A G-action on X is called proper if the map G×X → X ×X , (g, x) 7→(gx, x) is proper, i.e. the preimage of a compact subset is compact. Modify theargument in the proof of Theorem 5.4.5 (b) to show that the G-action on U(Σ) isproper whenever Σ is simplicial. (Hint: show that if sequences z (k) and g (k)z (k)have limits in U(Σ), then a subsequence of g (k) has a limit in G.)

5.4.16. Show that the quotient X/G of a locally compact Hausdorff space (e.g.a manifold)X by a properG-action is Hausdorff. Deduce that the quotient U(Σ)/Gcorresponding to a simplicial fan Σ is Hausdorff. This gives an alternative topolog-ical proof of Lemma 5.1.4 (separatedness of toric varieties) in the simplicial case.

5.4.17. Let Σ be a regular fan whose one-dimensional cones span NR. Observethat U(Σ) is 2-connected (see Exercise 4.7.11). Show by considering the exacthomotopy sequence of the principal G-bundle U(Σ) → VΣ that the nonsingular

5.5. HAMILTONIAN ACTIONS AND SYMPLECTIC REDUCTION 201

toric variety VΣ is simply connected, and H2(VΣ;Z) is naturally identified with thekernel of the map A : Zm → N . Hence H2(VΣ;Z) = Zm/N∗, which coincides withthe Picard group of VΣ, see [122, §3.4].

5.5. Hamiltonian actions and symplectic reduction

Here we describe projective toric manifolds as symplectic quotients of Hamil-tonian torus actions on Cm. This approach may be viewed as a symplectic geometryversion of the algebraic quotient construction from the previous section, althoughhistorically the symplectic construction preceded the algebraic one [11], [141].

Symplectic reduction. We briefly review the background material on sym-plectic geometry, referring the reader to monographs by Audin [11], Guillemin [141]and Guillemin–Ginzburg–Karshon [142] for further details.

A symplectic manifold is a pair (W,ω) consisting of a smooth manifold W anda closed differential 2-form ω which is nondegenerate at each point. The dimensionof a symplectic manifold W is necessarily even.

Assume now that a (compact) torus T acts on W preserving the symplecticform ω. We denote the Lie algebra of the torus T by t (since T is commutative, itsLie algebra is trivial, but the construction can be generalised to noncommutative Liegroups). Given an element v ∈ t, we denote by Xv the corresponding T -invariantvector field on W . The torus action is called Hamiltonian if the 1-form ω(Xv , · ) isexact for any v ∈ t. In other words, an action is Hamiltonian if for any v ∈ t thereexist a function Hv on W (called a Hamiltonian) satisfying the condition

ω(Xv , Y ) = dHv (Y )

for any vector field Y on W . The function Hv is defined up to addition of aconstant. Choose a basis ei in t and the corresponding Hamiltonians Hei

.Then the moment map

µ : W → t∗, (x, e i) 7→ Hei(x)

(where x ∈ W ) is defined. Observe that changing the Hamiltonians Heiby con-

stants results in shifting the image of µ by a vector in t∗. According to a theoremof Atiyah [8] and Guillemin–Sternberg [143], the image µ(W ) of the moment mapis convex, and if W is compact then µ(W ) is a convex polytope in t∗.

Example 5.5.1. The most basic example is W = Cm with symplectic form

ω = i

m∑

k=1

dzk ∧ dzk = 2

m∑

k=1

dxk ∧ dyk,

where zk = xk+iyk. The coordinatewise action of the torus Tm on Cm is Hamilton-ian. The moment map µ : Cm → Rm is given by µ(z1, . . . , zm) = (|z1|2, . . . , |zm|2)(an exercise). The image of the moment map µ is the positive orthant Rm> .

Construction 5.5.2 (symplectic reduction). Assume given a Hamiltonian ac-tion of a torus T on a symplectic manifold W . Assume further that the momentmap µ : W → t∗ is proper, i.e. µ−1(V ) is compact for any compact subset V ⊂ t∗

(this is always the case if W itself is compact). Let u ∈ t∗ be a regular value ofthe moment map, i.e. the differential TxW → t∗ is surjective for all x ∈ µ−1(u).Then the level set µ−1(u) is a smooth compact T -invariant submanifold in W .Furthermore, the T -action on µ−1(u) is almost free (an exercise).


Assume now that the T -action on µ−1(u) is free. The restriction of the sym-plectic form ω to µ−1(u) may be degenerate. However, the quotient manifoldµ−1(u)/T is endowed with is a unique symplectic form ω′ such that

p∗ω′ = i∗ω,

where i : µ−1(u)→W is the inclusion and p : µ−1(u)→ µ−1(u)/T the projection.We therefore obtain a new symplectic manifold (µ−1(u)/T, ω′) which is referred

to as the symplectic reduction, or the symplectic quotient of (W,ω) by T .The construction of symplectic reduction works also under milder assumptions

on the action (see [100] and more references there), but the generality describedhere will be enough for our purposes.

The toric case. The algebraic quotient construction describes a toric manifoldVΣ as a quotient of a noncompact set U(Σ) by a noncompact group G. Usingsymplectic reduction, the projective toric manifold VP corresponding to a simplelattice polytope P can be obtained as the quotient of a compact submanifold ZP ⊂U(ΣP ) by a free action of a compact torus.

Let Σ be a complete regular fan in NR∼= Rn with m one-dimensional cones gen-

erated by a1, . . . ,am. Consider the exact sequence of maximal compact subgroups(tori) corresponding to exact sequence of algebraic tori (5.2):

(5.9) 1 −→ K −→ TmexpA−→ TN −→ 1,

where TN = N⊗ZS ∼= Tn, expA : Tm → TN is the map of tori corresponding to themap of lattices A : Zm → N , ei 7→ ai, and K = KerA. The group K is isomorphicto Tm−n because Σ is complete and regular.

We let K ⊂ Tm act on Cm by restriction of the coordinatewise action of Tm.This K-action on Cm is also Hamiltonian, and the corresponding moment map isgiven by the composition

(5.10) µΣ : Cmµ−→ Rm −→ k∗,

where Rm → k∗ is the map of dual Lie algebras corresponding to the inclusionK → Tm. By choosing a basis in the weight lattice L ⊂ k∗ of the (m− n)-torus Kwe write the linear map Rm → k∗ by an integer (m − n) × m-matrix Γ = (γjk).Moment map (5.10) is then given by

(z1, . . . , zm) 7−→( m∑

k=1

γ1k|zk|2, . . . ,m∑

k=1

γm−n,k|zk|2),

and its level set µ−1Σ (δ) corresponding to a value δ = (δ1, . . . , δm−n) ∈ k∗ is the

intersection of m− n Hermitian quadrics in Cm:

(5.11)m∑

k=1

γjk|zk|2 = δj for j = 1, . . . ,m− n.

To apply the symplectic reduction we need to identify the regular values of themoment map µΣ. We recall from Section 2.1 that a polytope (1.1) is called Delzantif its normal fan is regular. If Σ = ΣP is the normal fan of a Delzant polytope P ,then the (m− n)×m-matrix Γ above is the one considered in Construction 1.2.1.Namely, the rows of Γ form a basis of linear dependencies between the vectors ai.


Given a polytope (1.1), we denote by ZP the intersection of quadrics (5.11)corresponding to δ = ΓbP , where bP = (b1, . . . , bm)t:

(5.12) ZP = µ−1Σ (ΓbP ) =

z ∈ Cm :

m∑

k=1

γjk(|zk|2−bk

)= 0 for j = 1, . . . ,m−n

.

Proposition 5.5.3. Assume that Σ = ΣP is the normal fan of a Delzantpolytope P given by (1.1). Then δ = ΓbP is a regular value of the moment map µΣ.

Proof. We only sketch the proof here; a more general statement will be provedas Theorem 6.3.1 in the next chapter. We need to check that this intersection isnondegenerate at each point z ∈ ZP , i.e. that ZP is a smooth submanifold in Cm.This means that the m − n gradient vectors of the left hand sides of quadraticequations (5.11) are linearly independent at each z ∈ ZP . This can be shown tobe equivalent to that the polytope P defined by (1.1) is simple.

As we shall see in Section 6.2, the manifold ZP is Tm-equivariantly homeomor-phic to the moment-angle manifold ZKP

(where KP is the nerve complex of P , orthe underlying simplicial complex of the normal fan ΣP ). Furthermore, we haveZP ⊂ U(ΣP ). (The reader may either wait until Chapter 6, or view these twostatements as exercises.)

We therefore may consider the symplectic quotient of Cm by K ∼= Tm−n, the2n-dimensional symplectic manifold µ−1

Σ (ΓbP )/K = ZP /K.

Theorem 5.5.4. Let P be a lattice Delzant polytope with the normal fan ΣP ,and let VP be the corresponding projective toric manifold. The inclusion ZP ⊂U(ΣP ) induces a diffeomorphism

ZP /K∼=−→ U(ΣP )/G = VP .

Therefore, any projective toric manifold VP is obtained as the symplectic quotientof Cm by an action of a torus K ∼= Tm−n.

Proof. We sketch the proof given in [11, Prop. VI.3.1.1]; a different proof ofa more general statement will be given in Section 6.6.

Consider the function

f : Cm → R, f(z ) = ‖µΣ(z )− ΓbP ‖2.It is nonnegative and minimal on the set ZP = µ−1

Σ (ΓbP ). The only criticalpoints of f in U(ΣP ) are z ∈ ZP . Hence, for any z ∈ U(ΣP ), the gradienttrajectory descending from z will reach a point in ZP . Furthermore, any gradienttrajectory is contained in a G-orbit. We therefore obtain that each G-orbit ofU(ΣP ) intersects ZP . Finally, it can be shown that each G-orbit intersects ZP ata unique K-orbit, i.e. for each z ∈ ZP we have that

G · z ∩ ZP = K · z .The statement follows.

The toric manifold VP therefore acquires a symplectic structure as the sym-plectic quotient µ−1

Σ (ΓbP )/K. On the other hand, the projective embedding ofVP defined by the lattice polytope P provides a symplectic form on VP by restric-tion of the standard symplectic form on the complex projective space. It can beshown [141, Appendix 2] that the diffeomorphism from Theorem 5.5.4 preserves


the cohomology class of the symplectic form, or equivalently, the two symplecticstructures on VP are TN -equivariantly symplectomorphic.

The symplectic quotient µ−1Σ (ΓbP )/K has a residual action of the quotient

n-torus TN = Tm/K, which is obviously Hamiltonian. This action is identified, viaTheorem 5.5.4, with the action of the maximal compact subgroup TN ⊂ C×

N on thetoric variety VP . We denote by µV : VP → t∗N the moment map for the Hamiltonianaction of TN on VP , where tN ∼= Rn is the Lie algebra of TN . It follows from (5.9)that t∗N embeds in Rm by the map A∗.

Proposition 5.5.5. The image of the moment map µV : VP → t∗N is the poly-tope P , up to translation.

Proof. Let ω be the standard symplectic form on Cm and µ : Cm → Rm themoment map for the standard action of Tm (see Example 5.5.1). Let p : ZP → VPbe the quotient projection by the action of K, and let i : ZP → Cm be the inclusion,so that the symplectic form ω′ on VP satisfies p∗ω′ = i∗ω. Let Hei

: Cm → R bethe Hamiltonian of the Tm-action on Cm corresponding to the ith basis vectorei (explicitly, Hei

(z ) = |zi|2), and let Hai: VP → R be the Hamiltonian of the

TN -action on VP corresponding to a i ∈ tN . Denote by Xeithe vector field on ZP

generated by e i, and denote by Yaithe vector field on VP generated by ai. Observe

that p∗Xei= Ya i

. For any vector field Z on ZP we have

dHei(Z) = i∗ω(Xei

, Z) = p∗ω′(Xei, Z)

= ω′(Ya i, p∗Z) = dHai

(p∗Z) = d(p∗Hai)(Z),

hence Hei= p∗Ha i

or Hei(z ) = Hai

(p (z )) up to constant. By definition of themoment map this implies that µV (VP ) ⊂ t∗N ⊂ Rm is identified with µ(ZP ) ⊂ Rm

up to shift by a vector in Rm. The inclusion t∗N ⊂ Rm is the map A∗, and µ(ZP ) =iA,b(P ) = A∗(P )+ b by definition of ZP , see (1.7) and (5.12). We therefore obtainthat there exists c ∈ Rm such that

A∗(µV (VP )) + c = A∗(P ) + b,

i.e. A∗(µV (VP )) and A∗(P ) differ by b − c ∈ A∗(t∗N ). Since A∗ is monomorphic,the result follows.

The symplectic quotient VP = µ−1Σ (ΓbP )/K with the Hamiltonian action of the

n-torus Tm/K is called the Hamiltonian toric manifold corresponding to a Delzantpolytope P .

According to the theorem of Delzant [93], any 2n-dimensional compact con-nected symplectic manifold W with an effective Hamiltonian action of an n-torus Tis equivariantly symplectomorphic to a Hamiltonian toric manifold VP , where P isthe image of the moment map µ : W → t∗ (whence the name ‘Delzant polytope’).

Remark. There is a canonical lattice in the dual Lie algebra t∗ (the weightlattice of the torus T ), and the moment polytope P = µ(W ) ⊂ t∗ satisfies theDelzant condition with respect to this lattice.

Example 5.5.6. Let P = ∆n be the standard simplex (see Example 1.2.2).The cones of the normal fan ΣP are spanned by the proper subsets of the set ofn + 1 vectors e1, . . . , en,−e1 − · · · − en. The groups G ∼= C× and K ∼= S1 arethe diagonal subgroups in (C×)n+1 and Tn+1 respectively, and U(ΣP ) = Cn+1 \0. The matrix Γ is a row of n + 1 units. The moment map (5.10) is given by


µΣ(z1, · · · , zn+1) = |z1|2 + . . .+ |zn+1|2. Since ΓbP = 1, the manifold ZP = µ−1P (1)

is the unit sphere S2n+1 ⊂ Cn+1, and S2n+1/K ∼= (Cn+1 \ 0)/G = VP is thecomplex projective space CPn.

Exercises.

5.5.7. The moment map µ : Cm → Rm for the coordinatewise action of Tm onCm is given by µ(z1, . . . , zm) = (|z1|2, . . . , |zm|2).

5.5.8. The T -action on µ−1(u) is almost free.

5.5.9. The level set ZP = µ−1Σ (ΓbP ) is Tm-equivariantly homeomorphic to the

moment-angle manifold ZKP.

5.5.10. Show that ZP ⊂ U(ΣP ).

CHAPTER 6

Geometric structures on moment-angle manifolds

In this chapter we study the geometry of moment-angle manifolds, in its convex,complex-analytic, symplectic and Lagrangian aspects.

As we have seen in Theorem 4.1.4, the moment-angle complex ZK correspond-ing to a triangulated sphere K is a topological manifold. Moment-angle manifoldscorresponding to simplicial polytopes or, more generally, complete simplicial fans,are smooth. In the polytopal case a smooth structure arises from the realisationof ZK by a nonsingular (transverse) intersection of Hermitian quadrics in Cm, sim-ilar to a level set of the moment map in the construction of symplectic quotients(see Section 5.5). The relationship between polytopes and systems of quadrics isdescribed in terms of Gale duality (see Sections 6.1 and 6.2).

Another way to give ZK a smooth structure is to realise it as the quotient of anopen subset in Cm (the complement U(K) of the coordinate subspace arrangementdefined by K) by an action of the multiplicative group Rm−n

> . As in the case of thequotient construction of toric varieties (Section 5.4), the quotient of a non-compactmanifold U(K) by the action of a non-compact group Rm−n

> is Hausdorff preciselywhen K is the underlying complex of a simplicial fan.

If m − n = 2ℓ is even, then the action of the real group Rm−n> on U(K) can

be turned into a holomorphic action of a complex (but not algebraic) group iso-morphic to Cℓ. In this way the moment-angle manifold ZK

∼= U(K)/Cℓ acquires acomplex-analytic structure. The resulting family of non-Kahler complex manifoldsgeneralises the well-known series as of Hopf and Calabi–Eckmann manifolds [70], aswell as LVM-manifolds (a class of complex structures on nonsingular intersectionsof Hermitian quadrics arising in holomorphic dynamics as transverse sets to certaincomplex foliations [198], [223]).

The intersections of Hermitian quadrics defining polytopal moment-angle man-ifolds can be also used to construct new families of Lagrangian submanifolds in Cm,CPm and toric varieties.

Particularly interesting examples of geometric structures (nonsingular in-tersections of quadrics, polytopal moment-angle manifolds, non-Kahler LVM-manifolds, Hamiltonian-minimal Lagrangian submanifolds) arise from Delzant poly-topes. These polytopes are abundant in toric topology (see Section 5.2).

6.1. Intersections of quadrics

Here we describe the correspondence between convex polyhedra and intersec-tions of quadrics. It will be used in the next section to define a smooth structureon moment-angle manifolds coming from polytopes.

From polyhedra to quadrics. The following construction originally ap-peared in [55, Construction 3.1.8] (see also [58, §3]):

207

208 6. GEOMETRIC STRUCTURES ON MOMENT-ANGLE MANIFOLDS

Construction 6.1.1. Consider a presentation of a convex polyhedron

(6.1) P = P (A, b) =x ∈ Rn : 〈a i, x 〉+ bi > 0 for i = 1, . . . ,m

.

Assume that a1, . . . ,am span Rn (i.e., P has a vertex) and recall the map

iA,b : Rn → Rm, iA,b(x ) = A∗x + b =

(〈a1, x 〉+ b1, . . . , 〈am, x 〉+ bm

)t

(see Construction 1.2.1). It embeds P into Rm> . We define the space ZA,b from thecommutative diagram

(6.2)

ZA,b iZ−−−−→ Cmy

yµ

PiA,b−−−−→ Rm>

where µ(z1, . . . , zm) = (|z1|2, . . . , |zm|2). The torus Tm acts on ZA,b with quotientP , and iZ is a Tm-equivariant embedding.

By replacing yk by |zk|2 in the equations defining the affine plane iA,b(Rn)(see (1.7)) we obtain that ZA,b embeds into Cm as the set of common zeros ofm− n quadratic equations (Hermitian quadrics):

(6.3) iZ(ZA,b) =z ∈ Cm :

m∑

k=1

γjk|zk|2 =

m∑

k=1

γjkbk, for 1 6 j 6 m− n.

The following property of the space ZA,b follows easily from its construction.

Proposition 6.1.2. Given a point z ∈ ZA,b, the jth coordinate of iZ(z) ∈ Cm

vanishes if and only if z projects onto a point x ∈ P such that x ∈ Fj .Theorem 6.1.3. The following conditions are equivalent:

(a) presentation (6.1) determined by the data (A, b) is generic;(b) the intersection of quadrics in (6.3) is nonempty and nonsingular, so thatZA,b is a smooth manifold of dimension m+ n.

Furthermore, under these conditions the embedding iZ : ZA,b → Cm has Tm-equivariantly framed normal bundle; a trivialisation is defined by a choice of matrixΓ = (γjk) in (1.7).

Proof. For simplicity we identify the space ZA,b with its embedded imageiZ(ZA,b) ⊂ Cm. We calculate the gradients of the m − n quadrics in (6.3) at apoint z = (x1, y1, . . . , xm, ym) ∈ ZA,b , where zk = xk + iyk:

(6.4) 2 (γj1x1, γj1y1, . . . , γjmxm, γjmym) , 1 6 j 6 m− n.These gradients form the rows of the (m− n)× 2m matrix 2Γ∆, where

∆ =

x1 y1 . . . 0 0...

.... . .

......

0 0 . . . xm ym

.

Let I = i1, . . . , ik = i : zi = 0 be the set of zero coordinates of z . Then therank of the gradient matrix 2Γ∆ at z is equal to the rank of the (m−n)× (m− k)matrix ΓI obtained by deleting the columns i1, . . . , ik from Γ .

Now let (6.1) be a generic presentation. By Proposition 6.1.2, a point z withzi1 = · · · = zik = 0 projects to a point in Fi1 ∩ · · · ∩ Fik 6= ∅. Hence the vectors

6.1. INTERSECTIONS OF QUADRICS 209

ai1 , . . . ,a ik are linearly independent. By Theorem 1.2.4, the rank of ΓI is m − n.Therefore, the intersection of quadrics (6.3) is nonsingular.

On the other hand, if (6.1) is not generic, then there is a point z ∈ ZA,b suchthat the vectors a i1 , . . . ,a ik : zi1 = · · · = zik = 0 are linearly dependent. ByTheorem 1.2.4, the columns of the corresponding matrix ΓI do not span Rm−n, soits rank is less than m−n and the intersection of quadrics (6.3) is degenerate at z .

The last statement follows from the fact that ZA,b is a nonsingular intersectionof quadratic surfaces, each of which is Tm-invariant.

From quadrics to polyhedra. This time we start with an intersection ofm− n Hermitian quadrics in Cm:

(6.5) ZΓ,δ =z = (z1, . . . , zm) ∈ Cm :

m∑

k=1

γjk|zk|2 = δj , for 1 6 j 6 m− n.

The coefficients of quadrics form an (m− n)×m-matrix Γ = (γjk), and we denoteits column vectors by γ1, . . . , γm. We also consider the column vector of the righthand sides, δ = (δ1, . . . , δm−n)

t ∈ Rm−n.These intersections of quadrics are considered up to linear equivalence, which

corresponds to applying a nondegenerate linear transformation of Rm−n to Γ and δ.Obviously, such a linear equivalence does not change the set ZΓ,δ.

We denote by R>〈γ1, . . . , γm〉 the cone generated by the vectors γ1, . . . , γm (i.e.,the set of linear combinations of these vectors with nonnegative coefficients).

A version of the following proposition appeared in [197], and the proof belowis a modification of the argument in [36, Lemma 0.3]. It allows us to determine thenondegeneracy of an intersection of quadrics directly from the data (Γ, δ):

Proposition 6.1.4. The intersection of quadrics ZΓ,δ given by (6.5) isnonempty and nonsingular if and only if the following two conditions are satis-fied:

(a) δ ∈ R>〈γ1, . . . , γm〉;(b) if δ ∈ R>〈γi1 , . . . γik〉, then k > m− n.

Under these conditions, ZΓ,δ is a smooth submanifold in Cm of dimension m+ n,and the vectors γ1, . . . , γm span Rm−n.

Proof. First assume that (a) and (b) are satisfied. Then (a) implies thatZΓ,δ 6= ∅. Let z ∈ ZΓ,δ. Then the rank of the matrix of gradients of (6.5) atz is equal to rkγk : zk 6= 0. Since z ∈ ZΓ,δ, the vector δ is in the cone gen-erated by those γk for which zk 6= 0. By the Caratheodory Theorem (see [325,§1.6]), δ belongs to the cone generated by some m − n of these vectors, that is,δ ∈ R>〈γk1 , . . . , γkm−n

〉, where zki 6= 0 for i = 1, . . . ,m − n. Moreover, the vec-tors γk1 , . . . , γkm−n

are linearly independent (otherwise, again by the CaratheodoryTheorem, we obtain a contradiction with (b)). This implies that the m− n gradi-ents of quadrics in (6.5) are linearly independent at z , and therefore ZΓ,δ is smoothand (m+ n)-dimensional.

To prove the other implication we observe that if (b) fails, that is, δ is in thecone generated by some m − n − 1 vectors of γ1, . . . , γm, then there is a pointz ∈ ZΓ,δ with at least n + 1 zero coordinates. The gradients of quadrics in (6.5)cannot be linearly independent at such z .


The torus Tm acts on ZΓ,δ, and the quotient ZΓ,δ/Tm is identified with the setof nonnegative solutions of the system of m− n linear equations

(6.6)

m∑

k=1

γkyk = δ.

This set can be described as a convex polyhedron P (A, b) given by (6.1), where(b1, . . . , bm) is any solution of (6.6) and the vectors a1, . . . ,am ∈ Rn form thetranspose of a basis of solutions of the homogeneous system

∑mk=1 γkyk = 0. We

refer to P (A, b) as the associated polyhedron of the intersection of quadrics ZΓ,δ. Ifthe vectors γ1, . . . , γm span Rm−n, then a1, . . . ,am span Rn. In this case the twovector configurations are Gale dual.

We summarise the results and constructions of this section as follows:

Theorem 6.1.5. A presentation of a polyhedron

P = P (A, b) =x ∈ Rn : 〈ai, x〉+ bi > 0 for i = 1, . . . ,m

(with a1, . . . ,am spanning Rn) defines an intersection of Hermitian quadrics

ZΓ,δ =z = (z1, . . . , zm) ∈ Cm :

m∑

k=1

γjk|zk|2 = δj for j = 1, . . . ,m− n.

(with γ1, . . . , γm spanning Rm−n) uniquely up to a linear isomorphism of Rm−n,and an intersection of quadrics ZΓ,δ defines a presentation P (A, b) uniquely up toan isomorphism of Rn.

The systems of vectors a1, . . . ,am ∈ Rn and γ1, . . . , γm ∈ Rm−n are Gale dual,and the vectors b ∈ Rm and δ ∈ Rm−n are related by the identity δ = Γb.

The intersection of quadrics ZΓ,δ is nonempty and nonsingular if and only ifthe presentation P (A, b) is generic.

Example 6.1.6 (m = n+ 1: one quadric). If presentation (6.1) is generic andP is bounded, then m > n+ 1. The case m = n + 1 corresponds to a simplex. IfP = P (A, b) is the standard simplex (see Example 1.2.2) we obtain

ZA,b = S2n+1 = z ∈ Cn+1 : |z1|2 + · · ·+ |zn+1|2 = 1.More generally, a presentation (6.1) with m = n + 1 and a1, . . . ,an spanning

Rn can be taken by an isomorphism of Rn to the form

P =x ∈ Rn : xi+bi > 0 for i = 1, . . . , n, and −c1x1−· · ·−cnxn+bn+1 > 0

.

We therefore have Γ = (c1 · · · cn 1), and ZA,b is given by the single equation

c1|z1|2 + · · ·+ cn|zn|2 + |zn+1|2 = c1b1 + · · ·+ cnbn + bn+1.

If the presentation is generic and bounded, then ZA,b is nonempty, nonsingular andbounded by Theorem 6.1.3. This implies that all ci and the right hand side aboveare positive, and ZA,b is an ellipsoid.

6.2. Moment-angle manifolds from polytopes

In this section we identify the polytopal moment-angle manifold ZKP(the

moment-angle complex corresponding to the nerve complex KP of a simple poly-tope P ) with the intersections of quadrics ZA,b (6.3).

A Tm-equivariant homeomorphism ZKP∼= ZA,b will be established using the

following construction of an identification space, which goes back to the work of

6.2. MOMENT-ANGLE MANIFOLDS FROM POLYTOPES 211

Vinberg [313] on Coxeter groups and was presented in the form described below inthe work of Davis and Januszkiewicz [90]. This was the first construction of whatlater became known as the moment-angle manifold.

Construction 6.2.1. Let [m] = 1, . . . ,m be the standard m-element set.For each I ⊂ [m] we consider the coordinate subtorus

TI = (t1, . . . , tm) ∈ Tm : tj = 1 for j /∈ I ⊂ Tm.

In particular, T∅ is the trivial subgroup 1 ⊂ Tm.Define the map R> × T→ C by (y, t) 7→ yt. Taking product we obtain a map

Rm> × Tm → Cm. The preimage of a point z ∈ Cm under this map is y × Tω(z ),where yi = |zi| for 1 6 i 6 m and ω(z ) = i : zi = 0 ⊂ [m] is the set of zerocoordinates of z . Therefore, Cm can be identified with the quotient space

(6.7) Rm> × Tm/∼ where (y , t1) ∼ (y , t2) if t−11 t2 ∈ Tω(y).

Given x ∈ P , set Ix = i ∈ [m] : x ∈ Fi (the set of facets containing x ).

Proposition 6.2.2. The space ZA,b given by intersection of quadrics (6.3)corresponding to a presentation P = P (A, b) is Tm-equivariantly homeomorphic tothe quotient

P × Tm/∼ where (x, t1) ∼ (x, t2) if t−11 t2 ∈ TIx .

Proof. Using (6.2), we identify the space ZA,b with iA,b(P )× Tm/∼ , where∼ is the equivalence relation from (6.7). A point x ∈ P is mapped by iA,b toy ∈ Rm> with Ix = ω(y).

An important corollary of this construction is that the topological type of theintersection of quadrics ZA,b depends only on the combinatorics of P :

Proposition 6.2.3. Assume given two generic presentations:

P =x ∈ Rn : (A∗x+ b)i > 0

and P ′ =

x ∈ Rn : (A′∗x+ b

′)i > 0

such that P and P ′ are combinatorially equivalent simple polytopes.

(a) If both presentations are irredundant, then the corresponding manifoldsZA,b and ZA′,b′ are Tm-equivariantly homeomorphic.

(b) If the second presentation is obtained from the first one by adding k re-dundant inequalities, then ZA′,b′ is homeomorphic to a product of ZA,band a k-torus T k.

Proof. (a) By Proposition 6.2.2, we have ZA,b ∼= P × Tm/∼ and ZA′,b′ ∼=P ′ × Tm/∼ . If both presentations are irredundant, then any Fi is a facet of P ,and the equivalence relation ∼ depends on the face structure of P only. There-fore, any homeomorphism P → P ′ preserving the face structure extends to a Tm-homeomorphism P × Tm/∼ → P ′ × Tm/∼ .

(b) Suppose the first presentation has m inequalities, and the second has m′

inequalities, so that m′ − m = k. Let J ⊂ [m′] be the subset corresponding tothe added redundant inequalities; we may assume that J = m+1, . . . ,m′. SinceFj = ∅ for any j ∈ J , we have Ix ∩ J = ∅ for any x ∈ P ′. Therefore, the

equivalence relation ∼ does not affect the factor TJ ⊂ Tm′

, and we have

ZA′,b′ ∼= P ′ × Tm′

/∼ ∼= (P × Tm/∼ )× TJ ∼= ZA,b × T k.


Remark. A Tm-homeomorphism in Proposition 6.2.3 (a) can be replaced bya Tm-diffeomorphism (with respect to the smooth structures of Theorem 6.1.3),but the proof is more technical. It follows from the fact that two simple polytopesare combinatorially equivalent if and only if they are diffeomorphic as ‘smoothmanifolds with corners’. For an alternative argument, see [36, Corollary 4.7].

Statement (a) remains valid without assuming that the presentation is genericor bounded, although ZA,b is not a manifold in this case.

Now we recall the moment-angle manifold ZKPcorresponding to the nerve

complex KP of a simple polytope P (see Section 4.1 and Example 2.2.4). Observethat in our formalism, redundant inequalities in a presentation of P correspond toghost vertices of KP .

Theorem 6.2.4. Let (6.1) be a generic bounded presentation, so that P =P (A, b) is a simple n-polytope. The moment-angle manifold ZKP

is Tm-equivari-antly homeomorphic to the intersection of quadrics ZA,b given by (6.3).

Proof. Recall from Construction 4.1.1 that the moment-angle complex ZKP

is defined from a diagram similar to (6.2), in which the bottom map is replaced bythe piecewise linear embedding cP : P → Im from Construction 2.9.7:

(6.8)

ZKP−−−−→ Dm

yyµ

PcP−−−−→ Im

As we have seen in Proposition 6.2.2, the intersection of quadrics ZA,b is Tm-homeomorphic to the identification space

P × Tm/∼ where (x , t1) ∼ (x , t2) if t−11 t2 ∈ TIx .

By restricting the equivalence relation (6.7) to Dm ⊂ Cm we obtain that

Dm ∼= Im × Tm/∼ where (y , t1) ∼ (y , t2) if t−11 t2 ∈ Tω(y).

As in the proof of Proposition 6.2.2, the space ZKPis identified with cP (P )×Tm/∼ .

A point x ∈ P is mapped by cP to y ∈ Im with Ix = ω(y) = i ∈ [m] : x ∈ Fi. Wetherefore obtain that both ZKP

and ZA,b are Tm-homeomorphic to P ×Tm/∼ .

Corollary 6.2.5. The moment-angle manifold ZKPcorresponding to the

nerve complex KP of a simple polytope P has a smooth structure in which theTm-action is smooth.

Definition 6.2.6. Given a bounded generic presentation (6.1) defining a simplepolytope P , we shall use the common notation ZP for both the moment-anglemanifold ZKP

and the intersection of quadrics (6.3). We refer to ZP as a polytopalmoment-angle manifold. We endow ZP with the smooth structure coming from thenonsingular intersection of quadrics.

Remark. As it is shown in [36, Corollary 4.7] a Tm-invariant smooth structureon ZP is unique. If the condition of the invariance under the torus action is dropped,then a smooth structure on ZP is not unique, as is shown by examples of odd-dimensional spheres and products of spheres. It would be interesting to relateexotic smooth structures with the construction of moment-angle complexes.

6.2. MOMENT-ANGLE MANIFOLDS FROM POLYTOPES 213

Remark. If the polytope P = P (A, b) is not simple, then moment-angle com-plex ZKP

corresponding to the nerve complex KP is not homeomorphic to the(singular) intersection of quadrics ZA,b . However, the two spaces are homotopyequivalent (see [12]).

An intersection of quadrics representing ZP can be chosen more canonically:

Proposition 6.2.7. The moment-angle manifold ZP is Tm-equivariantly dif-feomorphic to a nonsingular intersection of quadrics of the following form:

(6.9)

z ∈ Cm :

∑mk=1 |zk|2 = 1,∑m

k=1 gk|zk|2 = 0,

where (g1, . . . , gm) ⊂ Rm−n−1 is a combinatorial Gale diagram of P ∗.

Proof. It follows from Proposition 1.2.7 that ZP is given byz ∈ Cm : γ11|z1|2 + · · ·+ γ1m|zm|2 = c,

g1|z1|2 + · · ·+ gm|zm|2 = 0,

where γ1k and c are positive. Divide the first equation by c, and then replace each

zk by√

cγ1k

zk. As a result, each gk is multiplied by a positive number, so that

(g1, . . . , gm) remains to be a combinatorial Gale diagram for P ∗.

By adapting Proposition 6.1.4 to the special case of quadrics (6.9), we obtain

Proposition 6.2.8. The intersection of quadrics given by (6.9) is nonemptynonsingular if and only if the following two conditions are satisfied:

(a) 0 ∈ conv(g1, . . . , gm);(b) if 0 ∈ conv(gi1 , . . . , gik), then k > m− n.

Following [36], we refer to a nonsingular intersection (6.9) of m− n− 1 homo-geneous quadrics with a unit sphere in Cm as a link . We therefore obtain that theclass of links coincides with the class of polytopal moment-angle manifolds.

As we have seen in Example 6.1.6, the moment-angle manifold correspondingto an n-simplex is diffeomorphic to a sphere S2n+1. This is also the link of anempty system of homogeneous quadrics, corresponding to the case m = n+ 1.

Example 6.2.9 (m = n + 2: two quadrics). A polytope P defined by m =n+2 inequalities either is combinatorially equivalent to a product of two simplices(when there are no redundant inequalities), or is a simplex (when one inequality isredundant). In the case m = n+ 2 the link (6.9) has the form

z ∈ Cm : |z1|2 + · · ·+ |zm|2 = 1,

g1|z1|2 + · · ·+ gm|zm|2 = 0,

where gk ∈ R. Condition (b) of Proposition 6.2.8 implies that all gi are nonzero;assume that there are p positive and q = m−p negative numbers among them. Thencondition (a) implies that p > 0 and q > 0. Therefore, the link is the intersectionof the cone over a product of two ellipsoids of dimensions 2p− 1 and 2q− 1 (givenby the second quadric) with a unit sphere of dimension 2m− 1 (given by the firstquadric). Such a link is diffeomorphic to S2p−1 × S2q−1. The case p = 1 or q = 1corresponds to one redundant inequality. In the irredundant case (P is a product∆p−1 ×∆q−1, p, q > 1) we obtain that ZP ∼= S2p−1 × S2q−1.


The case of three quadrics was resolved by Lopez de Medrano [197] in 1989.Here is a reformulation of his result in terms of moment-angle manifolds:

Theorem 6.2.10. Let ZP be the moment-angle manifold given by a nonemptyand nodegenerate intersection of three quadrics (6.9), i.e. m = n+ 3. Then ZP isdiffeomorphic to a product of three odd-dimensional spheres or to a connected sumof products of spheres with two spheres in each product.

The proof of this theorem uses Gale duality, surgery theory and the h-cobordismTheorem. The original statement of [197] contained some restrictions on the typesof quadrics, which later were lifted in [127]. The moment-angle manifold ZP isdiffeomorphic to a product of three odd-dimensional spheres precisely when P iscombinatorially equivalent to a product of three simplices. In all other cases, themanifold ZP given by three quadrics is diffeomorphic to a connected sum of theform #m−2

k=3 (Sk × S2m−3−k)#qk . The numbers qk of products Sk × S2m−3−k in theconnected sum can be described explicitly in terms of the planar Gale diagram ofthe associated n-polytope P with m = n+ 3 facets.

Theorem 4.6.12 together with Theorem 6.2.10 and the previous examples givesa description of the topology of moment-angle manifolds ZP corresponding to thefollowing classes of simple n-polytopes P : dual stacked polytopes (including poly-gons), polytopes with m 6 n+ 3 facets, and products of them. More examples ofpolytopes P whose corresponding manifolds ZP are diffeomorphic to a connectedsum of sphere products were described in [127] (these include some dual cyclicpolytopes). In general, the topology of moment-angle manifolds is much more com-plicated than in these series of examples (see Section 4.9 where examples of ZPwith nontrivial Massey products were constructed). On the other hand, no otherexplicit topological types of moment-angle manifolds ZP are known. Furthermore,the following question remains open:

Problem 6.2.11. Does there exist a moment-angle manifold ZP decomposableinto nontrivial connected sum where one of the summands is diffeomorphic to aproduct of more than two spheres.

Here is an example illustrating how the structure of the moment-angle manifoldZP changes when one truncates P at a vertex:

Example 6.2.12. Consider the following presentation of a polygon:

P =(x1, x2) ∈ R2 : x1 > 0, x2 > 0,

− x1 + 1 + δ > 0, −x2 + 1 + ε > 0, −x1 − x2 + 1 > 0,

where δ, ε are parameters. First we fix some small positive δ and vary ε. If ε ispositive, then we get a presentation of a triangle with two redundant inequalities.The corresponding moment-angle manifold is diffeomorphic to S5×S1×S1. If ε isnegative, then we get a presentation of a quadrangle with one redundant inequality.The corresponding manifold is diffeomorphic to S3 × S3 × S1. We see that whenthe hyperplane −x2+1+ε = 0 crosses the vertex of the triangle, the correspondingmoment-angle manifold ZP undergoes a surgery turning S5×S1×S1 to S3×S3×S1.Now if we fix some small negative ε and decrease δ so that the hyperplane −x1 +1 + δ = 0 crosses the vertex of the quadrangle, then the manifold ZP undergoes amore complicated surgery turning the product S3 × S3 × S1 to the connected sum(S3 × S4)#5 (corresponding to the case n = 2,m = 5 in Theorem 4.6.12).

6.3. SYMPLECTIC REDUCTION AND MOMENT MAPS REVISITED 215

On the dual language of Gale diagrams, the surgeries described above happenwhen the origin crosses the “walls” inside the pentagonal Gale diagram correspond-ing to a presentation of a 2-polytope with 5 inequalities (see Figure 1.1). For moreinformation about surgeries of moment-angle manifolds and their relation to “wallcrossing”, see [55, §6.4], [36] and [127].

Exercises.

6.2.13. Let G ⊂ P be a face of codimension k in a simple n-polytope P ,let ZP be the corresponding moment-angle manifold with the quotient projectionp : ZP → P . Show that p−1(G) is a smooth submanifold of ZP of codimension 2k.Furthermore, p−1(G) is diffeomorphic to ZG × T ℓ, where ZG is the moment-anglemanifold corresponding to G and ℓ is the number of facets of P not containing G.

6.2.14. Let P = vt(I3) be the polytope obtained by truncating a 3-cube at avertex (see Construction 1.1.12). Write down intersection of quadrics (6.9) definingthe corresponding 10-dimensional moment-angle manifold ZP . Use Theorem 4.5.4or Theorem 4.5.7 to describe the cohomology ring H∗(ZP ). Deduce that ZP cannotbe diffeomorphic to a connected sum of sphere products (cf. [36, Example 11.5]).

6.2.15. Write down the quadratic equations defining the moment-angle mani-folds S5×S1×S1, S3×S3×S1 and (S3×S4)#5 from Example 6.2.12, and describeexplicitly the surgeries between them.

6.3. Symplectic reduction and moment maps revisited

As we have seen in Section 5.5, particular examples of polytopal moment-anglemanifolds ZP appear as level sets for the moment maps used in the constructionof Hamiltonian toric manifolds. In this case, the left hand sides of the equationsin (6.3) are quadratic Hamiltonians of a torus action on Cm. Here we investigatethe relationship between symplectic quotients of Cm and intersections of quadricsmore thoroughly. As a corollary, we obtain that any symplectic quotient of Cm bya torus action is a Hamiltonian toric manifold.

We want to study symplectic quotients of Cm by torus subgroups T ⊂ Tm.Such a subgroup of dimension m− n has the form

(6.10) TΓ =(e2πi〈γ1,ϕ〉, . . . , e2πi〈γm,ϕ〉

)∈ Tm

,

where ϕ ∈ Rm−n is an (m − n)-dimensional parameter, and Γ = (γ1, . . . , γm) isa set of m vectors in Rm−n. In order for TΓ to be a torus, the configuration ofvectors γ1, . . . , γm must be rational, i.e. their integer span L = Z〈γ1, . . . , γm〉 mustbe a full rank discrete subgroup (a lattice) in Rm−n. The lattice L is identifiedcanonically with Hom(TΓ , S1) and is called the weight lattice of torus (6.10). Let

L∗ = λ∗ ∈ Rm−n : 〈λ∗, λ〉 ∈ Z for all λ ∈ Lbe the dual lattice. We shall represent elements of TΓ by ϕ ∈ Rm−n occasionally,so that TΓ is identified with the quotient Rm−n/L∗.

The restricted action of TΓ ⊂ Tm on Cm is obviously Hamiltonian, and thecorresponding moment map is the composition

(6.11) µΓ : Cmµ−→ Rm −→ t∗Γ ,

where Rm → t∗Γ is the map of the dual Lie algebras corresponding to TΓ → Tm.The map Rm → t∗Γ takes the kth basis vector ek ∈ Rm to γk ∈ t∗Γ . By identifying


t∗Γ with Rm−n we write the map Rm → t∗Γ by the matrix Γ = (γjk), where γjk isthe jth coordinate of γk. The moment map (6.11) is then given by

(z1, . . . , zm) 7−→( m∑

k=1

γ1k|zk|2, . . . ,m∑

k=1

γm−n,k|zk|2).

Its level set µ−1Γ (δ) corresponding to a value δ = (δ1, . . . , δm−n)

t ∈ t∗Γ is exactly theintersection of quadrics ZΓ,δ given by (6.5).

To apply symplectic reduction we need to identify when the moment map µΓis proper, find its regular values δ, and finally identify when the action of TΓ onµ−1Γ (δ) = ZΓ,δ is free. In Theorem 6.3.1 below, all these conditions are expressed

in terms of the polyhedron P associated with ZΓ,δ as described in Section 6.1.It follows from Gale duality that γ1, . . . , γm span a lattice L in Rm−n if and

only if the dual configuration a1, . . . ,am spans a lattice N = Z〈a1, . . . ,am〉 in Rn.We refer to a presentation (6.1) as rational if Z〈a1, . . . ,am〉 is a lattice.

A polyhedron P is called Delzant if it has a vertex and there is a rationalpresentation (6.1) such that for any vertex x ∈ P the vectors a i normal to thefacets meeting at x form a basis of the lattice N = Z〈a1, . . . ,am〉. In the case whenP is bounded and irredundant we obtain the definition used before: a polytope Pis Delzant when its normal fan is regular.

Now let δ ∈ tΓ be a value of the moment map µΓ : Cm → t∗Γ , and µ−1Γ (δ) = ZΓ,δ

the corresponding level set, which is an intersection of quadrics (6.5). We associatewith ZΓ,δ a presentation (6.1) as described in Section 6.1 (see Theorem 6.1.5).

Theorem 6.3.1. Let TΓ ⊂ Tm be a torus subgroup (6.10), determined by arational configuration of vectors γ1, . . . , γm.

(a) The moment map µΓ : Cm → t∗Γ is proper if and only if its level set µ−1Γ (δ)

is bounded for some (and then for any) value δ ∈ t∗Γ . Equivalently, themap µΓ is proper if and only if the Gale dual configuration a1, . . . ,amsatisfies α1a1 + · · ·+ αmam = 0 for some positive numbers αk.

(b) δ ∈ t∗Γ is a regular value of µΓ if and only if the intersection of quadrics

µ−1Γ (δ) = ZΓ,δ is nonempty and nonsingular. Equivalently, δ is a regular

value if and only if the associated presentation P = P (A, b) is generic.(c) The action of TΓ on µ−1

Γ (δ) = ZΓ,δ is free if and only if the associatedpolyhedron P is Delzant.

Proof. (a) If µΓ is proper then µ−1Γ (δ) ⊂ t∗Γ is compact, so it is bounded.

Now assume that µ−1Γ (δ) = ZΓ,δ is bounded for some δ. Then the associated

polyhedron P is also bounded. By Corollary 1.2.8, this is equivalent to vanishing ofa positive linear combination of a1, . . . ,am. This condition is independent of δ, andwe conclude that µ−1

Γ (δ) is bounded for any δ. Let X ⊂ t∗Γ be a compact subset.

Since µ−1Γ (X) is closed, it is compact whenever it is bounded. By Proposition 1.2.7

we may assume that, for any δ ∈ X , the first quadric defining µ−1Γ (δ) = ZΓ,δ is

given by γ11|z1|2 + · · · + γ1m|zm|2 = δ1 with γ1k > 0. Let c = maxδ∈X δ1. Thenµ−1Γ (X) is contained in the bounded set

z ∈ Cm : γ11|z1|2 + · · ·+ γ1m|zm|2 6 cand is therefore bounded. Hence µ−1

Γ (X) is compact, and µΓ is proper.(b) The first statement is the definition of a regular value. The equivalent

statement is already proved as Theorem 6.1.3.

6.3. SYMPLECTIC REDUCTION AND MOMENT MAPS REVISITED 217

(c) We first need to identify the stabilisers of the TΓ -action on µ−1Γ (δ). Although

the fact that these stabilisers are finite for a regular value δ follows from the generalconstruction of symplectic reduction, we can prove this directly.

Given a point z = (z1, . . . , zm) ∈ ZΓ,δ, we define the sublattice

Lz = Z〈γi : zi 6= 0〉 ⊂ L = Z〈γ1, . . . , γm〉.Lemma 6.3.2. The stabiliser subgroup of z ∈ ZΓ,δ under the action of TΓ is

given by L∗z/L

∗. Furthermore, if ZΓ,δ is nonsingular, then all these stabilisers arefinite, i.e. the action of TΓ on ZΓ,δ is almost free.

Proof. An element (e2πi〈γ1,ϕ〉, . . . , e2πi〈γm,ϕ〉) ∈ TΓ fixes a point z ∈ ZΓ ifand only if e2πi〈γk,ϕ〉 = 1 whenever zk 6= 0. In other words, ϕ ∈ TΓ fixes z if andonly if 〈γk, ϕ〉 ∈ Z whenever zk 6= 0. The latter means that ϕ ∈ L∗

z . Since ϕ ∈ L∗

maps to 1 ∈ TΓ , the stabiliser of z is L∗z /L

∗.Assume now that ZΓ,δ is nonsingular. In order to see that L∗

z/L∗ is finite we

need to check that the sublattice Lz = Z〈γi : zi 6= 0〉 ⊂ L has full rank m − n.Indeed, rkγi : zi 6= 0 is the rank of the matrix of gradients of quadrics in (6.5)at z . Since ZΓ,δ is nonsingular, this rank is m− n, as needed.

Now we can finish the proof of Theorem 6.3.1 (c). Assume that P is Delzant.By Lemma 6.3.2, the TΓ -action on ZΓ,δ is free if and only if Lz = L for anyz ∈ ZΓ,δ. Let i : Zk → Zm be the inclusion of the coordinate sublattice spanned bythose ei for which zi = 0, and let p : Zm → Zm−k be the projection sending everysuch ei to zero. We also have the maps of lattices

Γ ∗ : L∗ → Zm, l 7→(〈γ1, l〉, . . . , 〈γm, l〉

), and A : Zm → N, ek 7→ ak.

Consider the diagram

(6.12)

0↓L∗

yΓ∗

0 −−−−→ Zki−−−−→ Zm

p−−−−→ Zm−k −−−−→ 0yAN↓0

in which the vertical and horizontal sequences are exact. Then the Delzant conditionis equivalent to that the composition A · i is split injective. The condition Lz = L isequivalent to that Γ ·p∗ is surjective, or p·Γ ∗ is split injective. These two conditionsare equivalent by Lemma 1.2.5.

In the case of polytopes we obtain the following version of Proposition 5.5.3:

Corollary 6.3.3. Let P = P (A, b) be a Delzant polytope, Γ = (γ1, . . . , γm) theGale dual configuration, and ZP the corresponding moment-angle manifold. Then

(a) δ = Γb is a regular value of the moment map µΓ : Cm → t∗Γ for theHamiltonian action of TΓ ⊂ Tm on Cm;

(b) ZP is the regular level set µ−1Γ (Γb);


(c) the action of TΓ on ZP is free.

In Section 5.5 we defined the Hamiltonian toric manifold VP correspondingto a Delzant polytope P as the symplectic quotient of Cm by the torus subgroupK ⊂ Tm determined by the normal fan of P . By comparing the vertical exactsequence in (6.12) with (5.9) we obtain that K = TΓ , and the quotient n-torusTm/TΓ acting on VP = ZP /TΓ is TN = N ⊗Z S = Rn/N .

Corollary 6.3.4. Any symplectic quotient of Cm by a torus subgroup T ⊂ Tm

is a Hamiltonian toric manifold.

Example 6.3.5. Consider the case m − n = 1, i.e. TΓ is 1-dimensional, andγk ∈ R. By Theorem 6.3.1 (a), the moment map µΓ is proper whenever

µ−1Γ (δ) = z ∈ Cm : γ1|z1|2 + · · ·+ γm|zm|2 = δ

is bounded for any δ ∈ R. By Theorem 6.3.1 (b), δ is a regular value whenever thequadratic hypersurface γ1|z1|2 + · · · + γm|zm|2 = δ is nonempty and nonsingular.These two conditions together imply that the hypersurface is an ellipsoid, and theassociated polyhedron is an n-simplex (see Example 6.1.6). By Lemma 6.3.2, theTΓ -action on µ−1

Γ (δ) is free if and only if Lz = L for any z ∈ µ−1Γ (δ). This means

that each γk generates the same lattice as the whole set γ1, . . . , γm, which impliesthat γ1 = · · · = γm. The Gale dual configuration satisfies a1 + · · ·+am = 0. ThenTΓ is the diagonal circle in Tm, the hypersurface µ−1

Γ (δ) = ZP is a sphere, andthe associated polytope P is the standard simplex up to shift and magnification bya positive factor δ. The Hamiltonian toric manifold VP = ZP /TΓ is the complexprojective space CPn.

6.4. Complex structures on intersections of quadrics

Bosio and Meersseman [36] identified polytopal moment-angle manifolds ZPwith a class of non-Kahler complex-analytic manifolds introduced in the works ofLopez de Medrano, Verjovsky and Meersseman (LVM-manifolds). This was thestarting point in the subsequent study of the complex geometry of moment-anglemanifolds. We review the construction of LVM-manifolds and its connection topolytopal moment-angle manifolds here.

The initial data of the construction of an LVM-manifold is a link of a homoge-neous system of quadrics similar to (6.9), but with complex coefficients:

(6.13) L =

z ∈ Cm :

∑mk=1 |zk|2 = 1,∑m

k=1 ζk|zk|2 = 0

,

where ζk ∈ Cs. We can obviously turn this link into the form (6.9) by identifyingCs with R2s in the standard way, so that each ζk becomes gk ∈ Rm−n−1 with n =m−2s−1. We assume that the link is nonsingular, i.e. the system of complex vectors(ζ1, . . . , ζm) (or the corresponding system of real vectors (g1, . . . , gm)) satisfies theconditions (a) and (b) of Proposition 6.2.8.

Now define the manifold N as the projectivisation of the intersection of homo-geneous quadrics in (6.13):

(6.14) N =z ∈ CPm−1 : ζ1|z1|2 + · · ·+ ζm|zm|2 = 0

, ζk ∈ Cs.

We therefore have a principal S1-bundle L → N .

6.4. COMPLEX STRUCTURES ON INTERSECTIONS OF QUADRICS 219

Theorem 6.4.1 (Meersseman [223]). The manifold N has a holomorphic atlasdescribing it as a compact complex manifold of complex dimension m− 1− s.

Sketch of proof. Consider a holomorphic action of Cs on Cm given by

(6.15)Cs × Cm −→ Cm

(w , z ) 7→(z1e

〈ζ1,w〉, . . . , zme〈ζm,w〉

),

where w = (w1, . . . , ws) ∈ Cs, and 〈ζk,w〉 = ζ1kw1 + · · ·+ ζskws.Let K be the simplicial complex consisting of zero-sets of points of the link L:

K = ω(z ) : z ∈ L.Observe that K = KP , where P is the simple polytope associated with the link L.Let U = U(K) be the corresponding subspace arrangement complement givenby (4.7.4). Note that Proposition 1.2.9 implies that U can be also defined as

U =(z1, . . . , zm) ∈ Cm : 0 ∈ conv(ζj : zj 6= 0)

.

An argument similar to the proof of Lemma 6.3.2 shows that the restrictionof the action (6.15) to U ⊂ Cm is free. Also, this restricted action of Cs on U isproper (we shall prove this in more general context in Theorem 6.6.3 below), so thequotient U/Cs is Hausdorff. Using a holomorphic atlas transverse to the orbits ofthe free action of Cs on the complex manifold U we obtain that the quotient U/Cs

has a structure of a complex manifold.On the other hand, it can be shown that the function |z1|2 + · · · + |zm|2 (the

square of the distance to the origin in Cm) has a unique minimum when restrictedto an orbit of the free action of Cs on U . The set of these minima (i.e. the set ofpoints closest to the origin in each orbit) can be described as

T =z ∈ Cm \0 : ζ1|z1|2 + · · ·+ ζm|zm|2 = 0

.

It follows that the quotient U/Cs can be identified with T , and therefore T acquiresa structure of a complex manifold of dimension m− s.

By projectivising the construction we identify N with the quotient of a comple-ment of coordinate subspace arrangement in CPm−1 (the projectivisation of U) by aholomorphic action of Cs. In this wayN becomes a compact complex manifold.

The manifold N with the complex structure of Theorem 6.4.1 is referred toas an LVM-manifold . These manifolds were described by Meersseman [223] as ageneralisation of the construction of Lopez de Medrano and Verjovsky [198].

Remark. The embedding of T in Cm and of N in CPm−1 given by (6.14) isnot holomorphic.

A polytopal moment-angle manifold ZP is diffeomorphic to a link (6.9), whichcan be turned into a complex link (6.13) whenever m + n is odd. It follows thatthe quotient ZP /S1 of an odd-dimensional moment-angle manifold has a complex-analytic structure as an LVM-manifold. By adding redundant inequalities andusing the S1-bundle L → N , Bosio–Meersseman observed that ZP or ZP × S1 hasa structure of an LVM-manifold, depending on whether m+ n is even or odd.

We first summarise the effects that a redundant inequality in (6.1) has ondifferent spaces appeared above:

Proposition 6.4.2. Assume that (6.1) is a generic presentation. The followingconditions are equivalent:


(a) 〈ai, x 〉+ bi > 0 is a redundant inequality in (6.1) (i.e. Fi = ∅);(b) ZP ⊂ z ∈ Cm : zi 6= 0;(c) i is a ghost vertex of KP ;(d) U(KP ) has a factor C× on the ith coordinate;(e) 0 /∈ conv(gk : k 6= i).

Proof. The equivalence of the first four conditions follows directly from thedefinitions. The equivalence (a)⇔(e) follows from Proposition 1.2.9.

Theorem 6.4.3 ([36]). Let ZP be the moment angle manifold correspondingto an n-dimensional simple polytope (6.1) defined by m inequalities.

(a) If m+ n is even then ZP has a complex structure as an LVM-manifold.(b) If m+n is odd then ZP ×S1 has a complex structure as an LVM-manifold.

Proof. (a) We add one redundant inequality of the form 1 > 0 to (6.1),and denote the resulting manifold of (6.2) by Z ′

P . We have Z ′P∼= ZP × S1. By

Proposition 6.2.7, ZP is diffeomorphic to a link given by (6.9). Then Z ′P is given

by the intersection of quadrics

z ∈ Cm+1 : |z1|2 + · · · + |zm|2 = 1,

g1|z1|2 + · · · + gm|zm|2 = 0,

|zm+1|2 = 1,

which is diffeomorphic to the link given by

z ∈ Cm+1 : |z1|2 + · · · + |zm|2 + |zm+1|2 = 1,

g1|z1|2 + · · · + gm|zm|2 = 0,

|z1|2 + · · · + |zm|2 − |zm+1|2 = 0.

If we denote by Γ ⋆ = (g1 . . . gm) the (m− n− 1)×m-matrix of coefficients of thehomogeneous quadrics for ZP , then the corresponding matrix for Z ′

P is

Γ ⋆′ =

(g1 · · · gm 01 · · · 1 −1

).

Its height m−n is even, so that we may think of its kth column as a complex vector

ζk (by identifying Rm−n with Cm−n

2 ), for k = 1, . . . ,m+ 1. Now define

(6.16) N ′ =z ∈ CPm : ζ1|z1|2 + · · ·+ ζm+1|zm+1|2 = 0

.

Then N ′ has a complex structure as an LVM-manifold by Theorem 6.4.1. On theother hand,

N ′ ∼= Z ′P /S

1 = (ZP × S1)/S1 ∼= ZP ,so that ZP also acquires a complex structure.

(b) The proof here is similar, but we have to add two redundant inequalities1 > 0 to (6.1). Then Z ′

P∼= ZP × S1 × S1 is given by

z ∈ Cm+2 : |z1|2 + · · · + |zm|2 + |zm+1|2 + |zm+2|2 = 1,

g1|z1|2 + · · · + gm|zm|2 = 0,

|z1|2 + · · · + |zm|2 − |zm+1|2 = 0,

|z1|2 + · · · + |zm|2 − |zm+2|2 = 0.

6.5. MOMENT-ANGLE MANIFOLDS FROM SIMPLICIAL FANS 221

The matrix of coefficients of the homogeneous quadrics is therefore

Γ ⋆′ =

g1 · · · gm 0 01 · · · 1 −1 01 · · · 1 0 −1

.

We think of its columns as a set of m+ 2 complex vectors ζ1, . . . , ζm+2, and define

(6.17) N ′ =z ∈ CPm+1 : ζ1|z1|2 + · · ·+ ζm+2|zm+2|2 = 0

.

Then N ′ has a complex structure as an LVM-manifold. On the other hand,

N ′ ∼= Z ′P /S

1 = (ZP × S1 × S1)/S1 ∼= ZP × S1,

and therefore ZP × S1 has a complex structure.

In the next two sections we describe a more direct method of endowing ZPwith a complex structure, without referring to projectivised quadrics and LVM-manifolds. This approach, developed in [262], works not only in the polytopalcase, but also for the moment-angle manifolds ZK corresponding to underlyingcomplexes K of complete simplicial fans.

6.5. Moment-angle manifolds from simplicial fans

LetK = KΣ be the underlying complex of a complete simplicial fan Σ, and U(K)the complement of the coordinate subspace arrangement (4.7.4) defined by K. Herewe shall identify the moment-angle manifold ZK with the quotient of U(K) by asmooth action of a non-compact group isomorphic to Rm−n, thereby defining asmooth structure on ZK. A modification of this construction will be used in thenext section to endow ZK with a complex structure.

Let Σ be a simplicial fan in NR∼= Rn with generators a1, . . . ,am. Recall that

the underlying simplicial complex K = KΣ is the collection of subsets I ⊂ [m] suchthat a i : i ∈ I spans a cone of Σ.

A simplicial fan Σ is therefore determined by two pieces of data:

– a simplicial complex K on [m];– a configuration of vectors a1, . . . ,am in Rn such that for any simplexI ∈ K the subset a i : i ∈ I is linearly independent.

Then for each I ∈ K we can define the simplicial cone σI spanned by a i with i ∈ I.The ‘bunch of cones’ σI : I ∈ K patches into a fan Σ whenever any two conesσI and σJ intersect in a common face (which has to be σI∩J). Equivalently, therelative interiors of cones σI are pairwise non-intersecting. Under this condition,we say that the data K;a1, . . . ,am define a fan Σ.

We do allow ghost vertices in K; they do not affect the fan Σ. The vectorai corresponding to a ghost vertex i can be zero as it does not correspond to aone-dimensional cone of Σ. This formalism will be important for the constructionof a complex structure on ZK; it was also used in [25] under the name triangulatedvector configurations.

Construction 6.5.1. For a set of vectors a1, . . . ,am, consider the linear map

(6.18) A : Rm → NR, ei 7→ a i,

where e1, . . . , em is the standard basis of Rm. Let

Rm> = (y1, . . . , ym) ∈ Rm : yi > 0


be the multiplicative group of m-tuples of positive real numbers, and define

(6.19)

R = exp(KerA) =(ey1, . . . , eym

): (y1, . . . , ym) ∈ KerA

=(t1, . . . , tm) ∈ Rm> :

m∏

i=1

t〈a i,u〉i = 1 for all u ∈ N∗

R

.

We let Rm> act on the complement U(K) ⊂ Cm by coordinatewise multiplica-tions and consider the restricted action of the subgroup R ⊂ Rm> . Recall that aG-action on a space X is proper if the group action map h : G × X → X × X ,(g, x) 7→ (gx, x) is proper (the preimage of a compact subset is compact).

Theorem 6.5.2 ([262]). Assume given data K;a1, . . . ,am satisfying the con-ditions above. Then

(a) the group R ∼= Rm−n given by (6.19) acts on U(K) freely;(b) if the data K;a1, . . . ,am define a simplicial fan Σ, then R acts on

U(K) properly, so the quotient U(K)/R is a smooth Hausdorff (m + n)-dimensional manifold;

(c) if the fan Σ is complete, then U(K)/R is homeomorphic to the moment-angle manifold ZK.

Therefore, ZK can be smoothed whenever K = KΣ for a complete simplicial fan Σ.

Proof. Statement (a) is proved in the same way as Proposition 5.4.6. Indeed,a point z ∈ U(K) has a nontrivial stabiliser with respect to the action of Rm> onlyif some of its coordinates vanish. These Rm> -stabilisers are of the form (R>, 1)I ,see (4.6), for some I ∈ K. The restriction of expA to any such (R>, 1)I is aninjection. Therefore, R = exp(KerA) intersects any Rm> -stabilisers only at theunit, which implies that the R-action on U(K) is free.

Let us prove (b) (compare the proof of Theorem 5.4.5 (b) and Exercise 5.4.15).Consider the map

h : R× U(K)→ U(K)× U(K), (g , z ) 7→ (rz , z ),

for r ∈ R, z ∈ U(K). Let V ⊂ U(K) × U(K) be a compact subset; we need toshow that h−1(V ) is compact. Since R × U(K) is metrisable, it suffices to checkthat any infinite sequence (r (k), z (k)) : k = 1, 2, . . . of points in h−1(V ) containsa converging subsequence. Since V ⊂ U(K) × U(K) is compact, by passing to asubsequence we may assume that the sequence

h(r (k), z (k)) = (r (k)z (k), z (k))has a limit in U(K)× U(K). We set w (k) = r (k)z (k), and assume that

w (k) → w = (w1, . . . , wm), z (k) → z = (z1, . . . , zm)

for some w , z ∈ U(K). We need to show that a subsequence of r (k) has limitin R. We write

r (k) =(g(k)1 , . . . , g(k)m

)=(eα

(k)1 , . . . , eα

(k)m

)∈ R ⊂ Rm> ,

α(k)j ∈ R. By passing to a subsequence we may assume that each sequence α(k)

j ,j = 1, . . . ,m, has a finite or infinite limit (including ±∞). Let

I+ = j : α(k)j → +∞ ⊂ [m], I− = j : α(k)

j → −∞ ⊂ [m].

6.5. MOMENT-ANGLE MANIFOLDS FROM SIMPLICIAL FANS 223

Since the sequences z (k), w (k) = r (k)z (k) converge to z ,w ∈ U(K) respectively,we have zj = 0 for j ∈ I+ and wj = 0 for j ∈ I−. Then it follows from thedecomposition U(K) =

⋃I∈K(C,C

×)I that I+ and I− are simplices of K. Letσ+, σ− be the corresponding cones of the simplicial fan Σ. Then σ+ ∩ σ− = 0by definition of a fan. By Lemma 2.1.2, there exists a linear function u ∈ N∗

R suchthat 〈u ,a〉 > 0 for any nonzero a ∈ σ+, and 〈u ,a〉 < 0 for any nonzero a ∈ σ−.Since r (k) ∈ R, it follows from (6.19) that

(6.20)

m∑

j=1

α(k)j 〈u ,aj〉 = 0.

This implies that both I+ and I− are empty, as otherwise the latter sum tends to

infinity. Thus, each sequence α(k)j has a finite limit αj , and a subsequence of

r (k) converges to (eα1 , . . . , eαm). Passing to the limit in (6.20) we obtain that(eα1 , . . . , eαm) ∈ R. This proves the properness of the action. Since the Lie groupR acts smoothly, freely and properly on the smooth manifold U(K), the orbit spaceU(K)/R is Hausdorff and smooth by the standard result [193, Theorem 9.16].

In the case of complete fan it is possible to construct a smooth atlas on U(K)/Rexplicitly. To do this, it is convenient to pre-factorise everything by the action ofTm, as in the proof of Theorem 4.7.5. We have

U(K)/Tm = (R>,R>)K =

⋃

I∈K

(R>,R>)I .

Since the fan Σ is complete, we may take the union above only over n-elementsimplices I = i1, . . . , in ∈ K. Consider one such simplex I; the generators of thecorresponding n-dimensional cone σ ∈ Σ are a i1 , . . . ,a in . Let u1, . . . ,un be thedual basis of N∗

R (which is a generator set of the dual cone σv). Then we have〈a ik ,uj〉 = δkj . Now consider the map

pI : (R>,R>)I → Rn>

(y1, . . . , ym) 7→( m∏

i=1

y〈ai,u1〉i , . . . ,

m∏

i=1

y〈ai,un〉i

),

where we set 00 = 1. Note that zero does not occur with a negative exponent inthe right hand side, hence pI is well defined as a continuous map. Each (R>,R>)I

is R-invariant, and it follows from (6.19) that pI induces an injective map

qI : (R>,R>)I/R→ Rn>.

This map is also surjective since every (x1, . . . , xn) ∈ Rn> is covered by (y1, . . . , ym)

where yij = xj for 1 6 j 6 n and yk = 1 for k /∈ i1, . . . , in. Hence, qI is ahomeomorphism. It is covered by a Tm-equivariant homeomorphism

qI : (C,C×)I/R→ Cn × Tm−n,

where Cn is identified with the quotient Rn> × Tn/∼ , see (6.7). Since U(K)/R is

covered by open subsets (C,C×)I/R, and Cn × Tm−n embeds as an open subsetin Rm+n, the set of homeomorphisms qI : I ∈ K provides an atlas for U(K)/R.The change of coordinates transformations qJq

−1I : Cn × Tm−n → Cn × Tm−n are

smooth by inspection; thus U(K)/R is a smooth manifold.


Remark. The set of homeomorphisms qI : (R>,R>)I/R → Rn> defines an

atlas for the smooth manifold with corners ZK/Tm. If K = KP for a simple polytopeP , then this smooth structure with corners coincides with that of P .

It remains to prove statement (c), that is, identify U(K)/R with ZK. If Xis a Hausdorff locally compact space with a proper G-action, and Y ⊂ X a com-pact subspace which intersects every G-orbit at a single point, then Y is homeo-morphic to the orbit space X/G. Therefore, we need to verify that each R-orbitintersects ZK ⊂ U(K) at a single point. We first prove that the R-orbit of anyy ∈ U(K)/Tm = (R>,R>)K intersects ZK/Tm at a single point. For this we usethe cubical decomposition cc(K) = (I, 1)K of ZK/Tm, see Example 4.2.6.2.

Assume first that y ∈ Rm> . The R-action on Rm> is obtained by exponentiatingthe linear action of KerA on Rm. Consider the subset (R6, 0)

K ⊂ Rm, where R6

denotes the set of nonpositive reals. It is taken by the exponential map exp: Rm →Rm> homeomorphically onto cc(K) = ((0, 1], 1)K ⊂ Rm> , where (0, 1] is denotes thesemi-interval y ∈ R : 0 < y 6 1. The map

(6.21) A : (R6, 0)K → NR

takes every (R6, 0)I to −σ, where σ ∈ Σ is the cone corresponding to I ∈ K. Since

Σ is complete, map (6.21) is one-to-one.The orbit of y under the action of R consists of points w ∈ Rm> such that

expAw = expAy . Since Ay ∈ NR and map (6.21) is one-to-one, there is a uniquepoint y ′ ∈ (R6, 0)

K such that Ay ′ = Ay . Since expAy ′ ⊂ cc(K), the R-orbit ofy intersects cc(K) and therefore cc(K) at a unique point.

Now let y ∈ (R>,R>)K be an arbitrary point. Let ω(y) ∈ K be the set ofzero coordinates of y , and let σ ∈ Σ be the cone corresponding to ω(y). Thecones containing σ constitute a fan stσ (called the star of σ) in the quotient spaceNR/R〈a i : i ∈ ω(y)〉. The underlying simplicial complex of stσ is the link lkK ω(y)of ω(y) in K. Now observe that the action of R on the set

(y1, . . . , ym) ∈ (R>,R>)K : yi = 0 for i ∈ ω(y) ∼= (R>,R>)

lkω(y)

coincides with the action of the group Rstσ (defined by the fan stσ). Now we canrepeat the above arguments for the complete fan stσ and the action of Rstσ on(R>,R>)lkω(y). As a result, we obtain that every R-orbit intersects cc(K) at aunique point.

To finish the proof of (c) we consider the commutative diagram

ZK −−−−→ U(K)y

yπ

cc(K) −−−−→ (R>,R>)K

where the horizontal arrows are embeddings and the vertical ones are projectionsonto the quotients of Tm-actions. Note that the projection π commutes with the R-actions on U(K) and (R>,R>)K, and the subgroups R and Tm of (C×)m intersecttrivially. It follows that every R-orbit intersects the full preimage π−1(cc(K)) = ZK

at a unique point. Indeed, assume that z and rz are in ZK for some z ∈ U(K)and r ∈ R. Then π(z ) and π(rz ) = rπ(z ) are in cc(K), which implies that π(z ) =π(rz ). Hence, z = trz for some t ∈ Tm. We may assume that z ∈ (C×)m, so thatthe action of both R and Tm is free (otherwise consider the action on U(lkω(z ))).It follows that tr = 1, which implies that r = 1, since R ∩ Tm = 1.

6.6. COMPLEX STRUCTURES ON MOMENT-ANGLE MANIFOLDS 225

We do not know if Theorem 6.5.2 generalises to other sphere triangulations:

Problem 6.5.3. Describe the class of sphere triangulations K for which themoment-angle manifold ZK admits a smooth structure.

Remark. Even if ZK admits a smooth structure for some simplicial complexesK not arising from fans, such a structure does not come from a quotient U(K)/Rdetermined by data K;a1, . . . ,am. Like in the toric case (see Section 5.4), theR-action on U(K) is proper and the quotient U(K)/R is Hausdorff precisely whenK;a1, . . . ,am defines a fan, i.e. the simplicial cones generated by any two subsetsai : i ∈ I and aj : j ∈ J with I, J ∈ K can be separated by a hyperplane. Thisobservation is originally due to Bosio [35], see also [6, §II.3] and [25].

6.6. Complex structures on moment-angle manifolds

Let ZK be the moment-angle manifold corresponding to a complete simplicialfan Σ defined by data K;a1, . . . ,am. We assume that the dimension m + n ofZK is even, and set m − n = 2ℓ. This can always be achieved by adding a ghostvertex with any corresponding vector to our data K;a1, . . . ,am; topologically thisresults in multiplying ZK by a circle. Here we show that ZK admits a structureof a complex manifold. The idea is to replace the action of R ∼= Rm−n on U(K)(whose quotient is ZK) by a holomorphic action of C

m−n2 on the same space.

We identify Cm (as a real vector space) with R2m using the map

(z1, . . . , zm) 7→ (x1, y1, . . . , xm, ym),

where zk = xk + iyk, and consider the R-linear map

Re: Cm → Rm, (z1, . . . , zm) 7→ (x1, . . . , xm).

In order to obtain a complex structure on the quotient ZK∼= U(K)/R we

replace the action of R by the action of a holomorphic subgroup C ⊂ (C×)m bymeans of the following construction.

Construction 6.6.1. Let a1, . . . ,am be a configuration of vectors that spanNR∼= Rn, and assume that m− n = 2ℓ. Some of the ai’s may be zero. Recall the

map A : Rm → NR, ei 7→ a i.We choose a complex ℓ-dimensional subspace in Cm which projects isomorphi-

cally onto the real (m− n)-dimensional subspace KerA ⊂ Rm. More precisely, letc ∼= Cℓ, and choose a linear map Ψ : c→ Cm satisfying the two conditions:

(a) the composite map cΨ−→ Cm

Re−→ Rm is a monomorphism;

(b) the composite map cΨ−→ Cm

Re−→ RmA−→ NR is zero.

These two conditions are equivalent to the following:

(a’) Ψ(c) ∩ Ψ(c)= 0;(b’) Ψ(c) ⊂ Ker(AC : Cm → NC),

where Ψ(c) is the complex conjugate space and AC : Cm → NC is the complexifica-tion of the real map A : Rm → NR. Consider the following commutative diagram:

(6.22)

cΨ−−−−→ Cm

Re−−−−→ RmA−−−−→ NRyexp

yexp

(C×)m| ·|−−−−→ Rm>


where the vertical arrows are the componentwise exponential maps, and | · | denotesthe map (z1, . . . , zm) 7→ (|z1|, . . . , |zm|). Now set

(6.23) CΨ = expΨ(c) =(e〈ψ1,w〉, . . . , e〈ψm,w〉

)∈ (C×)m

where w ∈ c and ψk ∈ c∗ is given by the kth coordinate projection cΨ−→ Cm → C.

Then CΨ ∼= Cℓ is a complex-analytic (but not algebraic) subgroup in (C×)m, andtherefore there is a holomorphic action of CΨ on Cm and U(K) by restriction.

Example 6.6.2. Let a1, . . . ,am be the configuration of m = 2ℓ zero vectors.We supplement it by the empty simplicial complex K on [m] (with m ghost vertices),so that the data K;a1, . . . ,am define a complete fan in 0-dimensional space.Then A : Rm → R0 is a zero map, and condition (b) of Construction 6.6.1 is void.

Condition (a) means that cΨ−→ C2ℓ Re−→ R2ℓ is an isomorphism of real spaces.

Consider the quotient (C×)m/CΨ (note that U(K) = (C×)m in our case). Theexponential map Cm → (C×)m identifies (C×)m with the quotient of Cm by theimaginary lattice Γ = Z〈2πie1, . . . , 2πiem〉. Condition (a) implies that the pro-jection p : Cm → Cm/Ψ(c) is nondegenerate on the imaginary subspace of Cm. Inparticular, p (Γ) is a lattice of rank m = 2ℓ in Cm/Ψ(c) ∼= Cℓ. Therefore,

(C×)m/CΨ ∼=(Cm/Γ

)/Ψ(c) =

(Cm/Ψ(c)

)/p (Γ) ∼= Cℓ/p (Γ)

is a complex compact ℓ-dimensional torus.Any complex torus can be obtained in this way. Indeed, let Ψ : c → Cm be

given by an 2ℓ × ℓ-matrix

(−BI

)where I is the unit matrix and B is a square

matrix of size ℓ. Then p : Cm → Cm/Ψ(c) is given by the matrix (I B) in ap-propriate bases, and (C×)m/CΨ is isomorphic to the quotient of Cℓ by the latticeZ〈e1, . . . , eℓ, b1, . . . , bℓ〉, where bk is the kth column of B. (Condition (b) impliesthat the imaginary part of B is nondegenerate.)

For example, if ℓ = 1, then Ψ : C → C2 is given by w 7→ (βw,w) for someβ ∈ C, so that subgroup (6.23) is

CΨ = (eβw, ew) ⊂ (C×)2.

Condition (a) implies that β /∈ R. Then expΨ : C→ (C×)2 is an embedding, and

(C×)2/CΨ ∼= C/(Z⊕ βZ) = T 1C(β)

is a complex 1-dimensional torus with lattice parameter β ∈ C.

Theorem 6.6.3 ([262]). Assume that data K;a1, . . . ,am define a completefan Σ in NR

∼= Rn, and m − n = 2ℓ. Let CΨ ∼= Cℓ be the group given by (6.23).Then

(a) the holomorphic action of CΨ on U(K) is free and proper, and the quotientU(K)/CΨ has a structure of a compact complex manifold;

(b) U(K)/CΨ is diffeomorphic to the moment-angle manifold ZK.

Therefore, ZK has a complex structure, in which each element of Tm acts by aholomorphic transformation.

Remark. A result similar to Theorem 6.6.3 was obtained by Tambour [301].The approach of Tambour was somewhat different; he constructed complex struc-tures on manifolds ZK arising from rationally starshaped spheres K (underlying

6.6. COMPLEX STRUCTURES ON MOMENT-ANGLE MANIFOLDS 227

complexes of complete rational simplicial fans) by relating them to a class of gen-eralised LVM-manifolds described by Bosio in [35].

Proof of Theorem 6.6.3. We first prove statement (a). The stabilisers ofthe (C×)m-action on U(K) are of the form (C×, 1)I for I ∈ K. In order to showthat CΨ ⊂ (C×)m acts freely we need to check that CΨ has trivial intersectionwith any stabiliser of the (C×)m-action. Since CΨ embeds into Rm> by (6.22), itenough to check that the image of CΨ in Rm> intersects the image of (C×, 1)I in Rm>trivially. The former image is R and the latter image is (R>, 1)I ; the triviality oftheir intersection follows from Theorem 6.5.2 (a).

Now we prove the properness of this action. Consider the projection π : U(K)→(R>,R>)K onto the quotient of the Tm-action, and the commutative square

CΨ × U(K) hC−−−−→ U(K) × U(K)yf×π

yπ×π

R× (R>,R>)KhR−−−−→ (R>,R>)K × (R>,R>)K

where hC and hR denote the group action maps, and f : CΨ → R is the isomorphismgiven by the restriction of | · | : (C×)m → Rm> . The preimage h−1

C (V ) of a compact

subset V ∈ U(K) × U(K) is a closed subset in W = (f × π)−1 h−1R (π × π)(V ).

The image (π × π)(V ) is compact, the action of R on (R>,R>)K is proper byTheorem 6.5.2 (b), and the map f × π is proper as the quotient projection for acompact group action. Hence, W is a compact subset in CΨ × U(K), and h−1

C (V )is compact as a closed subset in W .

The group CΨ ∼= Cl acts holomorphically, freely and properly on the complexmanifold U(K), therefore the quotient manifold U(K)/CΨ has a complex structure.

As in the proof of Theorem 6.5.2, it is possible to describe a holomorphic atlas ofU(K)/CΨ . Since the action of CΨ on the quotient U(K)/Tm = (R>,R>)

K coincideswith the action of R on the same space, the quotient of U(K)/CΨ by the action ofTm has exactly the same structure of a smooth manifold with corners as the quotientof U(K)/R by Tm (see the proof of Theorem 6.5.2). This structure is determinedby the atlas qI : (R>,R>)I/R→ Rn>, which lifts to a covering of U(K)/CΨ by the

open subsets (C,C×)I/CΨ . For any I ∈ K, the subset (C,T)I ⊂ (C,C×)I intersectseach orbit of the CΨ -action on (C,C×)I transversely at a single point. Therefore,every (C,C×)I/CΨ ∼= (C,T)I acquires a structure of a complex manifold. Since(C,C×)I ∼= Cn × (C×)m−n, and the action of CΨ on the (C×)m−n factor is free,the complex manifold (C,C×)I/CΨ is the total space of a holomorphic Cn-bundleover the complex torus (C×)m−n/CΨ (see Example 6.6.2). Writing trivialisationsof these Cn-bundles for every I, we obtain a holomorphic atlas for U(K)/CΨ .

The proof of statement (b) follows the lines of the proof of Theorem 6.5.2 (b).We need to show that each CΨ -orbit intersects ZK ⊂ U(K) at a single point. Firstwe show that the CΨ -orbit of any point in U(K)/Tm intersects ZK/Tm = cc(K) ata single point; this follows from the fact that the actions of CΨ and R coincide onU(K)/Tm. Then we show that each CΨ -orbit intersects the preimage π−1(cc(K)) ata single point, using the fact that CΨ and Tm have trivial intersection in (C×)m.

Example 6.6.4 (Hopf manifold). Let a1, . . . ,an+1 be a set of vectors whichspan NR

∼= Rn and satisfy a linear relation λ1a1 + · · · + λn+1an+1 = 0 with allλk > 0. Let Σ be the complete simplicial fan in NR whose cones are generated by


all proper subsets of a1, . . . ,an+1. To make m − n even we add one more ghostvector an+2. Hence m = n + 2, ℓ = 1, and we have one more linear relationµ1a1 + · · · + µn+1an+1 + an+2 = 0 with µk ∈ R. The subspace KerA ⊂ Rn+2 isspanned by (λ1, . . . , λn+1, 0) and (µ1, . . . , µn+1, 1).

Then K = KΣ is the boundary of an n-dimensional simplex with n+1 verticesand one ghost vertex, ZK

∼= S2n+1 × S1, and U(K) = (Cn+1 \ 0)× C×.Conditions (a) and (b) of Construction 6.6.1 imply that CΨ is a 1-dimensional

subgroup in (C×)m given in appropriate coordinates by

CΨ =(eζ1w, . . . , eζn+1w, ew) : w ∈ C

⊂ (C×)m,

where ζk = µk+αλk for some α ∈ C\R. By changing the basis of KerA if necessary,we may assume that α = i. The moment-angle manifold ZK

∼= S2n+1×S1 acquiresa complex structure as the quotient U(K)/CΨ :

(Cn+1 \ 0

)× C×

/ (z1, . . . , zn+1, t)∼ (eζ1wz1, . . . , e

ζn+1wzn+1, ewt)

∼=(Cn+1 \ 0

)/ (z1, . . . , zn+1)∼ (e2πiζ1z1, . . . , e

2πiζn+1zn+1),

where z ∈ Cn+1 \ 0, t ∈ C×. The latter is the quotient of Cn+1 \ 0 by adiagonalisable action of Z. It is known as a Hopf manifold. For n = 0 we obtainthe complex torus of Example 6.6.2.

Theorem 6.6.3 can be generalised to the quotients of ZK by freely acting sub-groups H ⊂ Tm, or partial quotients of ZK in the sense of [55, §7.5]. These includeboth toric manifolds and LVM-manifolds:

Construction 6.6.5. Let Σ be a complete simplicial fan in NR defined bydata K;a1, . . . ,am, and let H ⊂ Tm be a subgroup which acts freely on thecorresponding moment-angle manifold ZK. Then H is a product of a torus and afinite group, and h = dimH 6 m−n by Proposition 5.4.6 (H must intersect triviallywith an n-dimensional coordinate subtorus in Tm). Under an additional assumptionon H , we shall define a holomorphic subgroupD in (C×)m and introduce a complexstructure on ZK/H by identifying it with the quotient U(K)/D.

The additional assumption is the compatibility with the fan data. Recall themap A : Rm → NR, ei 7→ a i, and let h ⊂ Rm be the Lie algebra of H ⊂ Tm. Weassume that h ⊂ KerA. We also assume that 2ℓ = m− n− h is even (this can besatisfied by adding a zero vector to a1, . . . ,am). Let T = Tm/H be the quotienttorus, t its Lie algebra, and ρ : Rm → t the map of Lie algebras corresponding tothe quotient projection Tm → T .

Let c ∼= Cℓ, and choose a linear map Ω : c→ Cm satisfying the two conditions:

(a) the composite map cΩ−→ Cm

Re−→ Rmρ−→ t is a monomorphism;

(b) the composite map cΩ−→ Cm

Re−→ RmA−→ NR is zero.

Equivalently, choose a complex subspace c ⊂ tC such that the composite map

c→ tCRe−→ t is a monomorphism.

As in Construction 6.6.1, expΩ(c) ⊂ (C×)m is a holomorphic subgroup iso-morphic to Cℓ. Let HC ⊂ (C×)m be the complexification of H (it is a product of(C×)h and a finite group). It follows from (a) that the subgroups HC and expΩ(c)intersect trivially in (C×)m. We can define a complex (h+ ℓ)-dimensional subgroup

(6.24) DH,Ω = HC × expΩ(c) ⊂ (C×)m.

6.7. HOLOMORPHIC PRINCIPAL BUNDLES AND DOLBEAULT COHOMOLOGY 229

Theorem 6.6.6 ([262, Theorem 3.7]). Let Σ, K and DH,Ω be as above. Then

(a) the holomorphic action of the group DH,Ω on U(K) is free and proper, andthe quotient U(K)/DH,Ω has a structure of a compact complex manifoldof complex dimension m− h− ℓ;

(b) there is a diffeomorphism between U(K)/DH,Ω and ZK/H defining a com-plex structure on the quotient ZK/H, in which each element of T = Tm/Hacts by a holomorphic transformation.

The proof is similar to that of Theorem 6.6.3 and is omitted.

Example 6.6.7.1. If H is trivial (h = 0) then we obtain Theorem 6.6.3.2. Let H be the diagonal circle in Tm. The condition h ⊂ KerAR implies that

the vectors a1, . . . ,am sum up to zero, which can always be achieved by rescalingthem (as Σ is a complete fan). As the result, we obtain a complex structure on thequotient ZK/S

1 by the diagonal circle in Tm, provided that m− n is odd. In thepolytopal case K = KP , the quotient ZK/S

1 embeds into Cm \ 0/C× = CPm−1

as an intersection of homogeneous quadrics (6.14), and the complex structure onZK/S

1 coincides with that of an LVM-manifold , see Section 6.4.3. Let h = dimH = m − n. Then h = KerA. Since h is the Lie algebra of a

torus, the (m− n)-dimensional subspace KerA ⊂ Rm is rational. By Gale duality,this implies that the fan Σ is also rational. We have ℓ = 0, DH,Ω = HC

∼= (C×)m−n

and U(K)/HC = ZK/H is the toric variety corresponding to Σ.

An effective action of T k on an m-dimensional manifold M is called maximal ifthere exists a point x ∈M whose stabiliser has dimension m− k; the two extremecases are the free action of a torus on itself and the half-dimensional torus actionon a toric manifold. As it is shown by Ishida [169], any compact complex man-ifold with a maximal effective holomorphic action of a torus is biholomorphic toa quotient ZK/H of a moment-angle manifold with a complex structure describedby Theorem 6.6.6. The argument of [169] recovering a fan Σ from a maximalholomorphic torus action builds up on the works [170] and [171], where the re-sult was proved in particular cases. The main result of [171] provides a purelycomplex-analytic description of toric manifolds VΣ:

Theorem 6.6.8 ([171, Theorem 1]). Let M be a compact connected complexmanifold of complex dimension n, equipped with an effective action of T n by holo-morphic transformations. If the action has fixed points, then there exists a completeregular fan Σ and a T n-equivariant biholomorphism of VΣ with M .

6.7. Holomorphic principal bundles and Dolbeault cohomology

In the case of rational simplicial normal fans ΣP a construction of Meersseman–Verjovsky [224] identifies the corresponding projective toric variety VP as the baseof a holomorphic principal Seifert fibration, whose total space is the moment-anglemanifold ZP equipped with a complex structure of an LVM-manifold, and fibreis a compact complex torus of complex dimension ℓ = m−n

2 . (Seifert fibrationsare generalisations of holomorphic fibre bundles to the case when the base is anorbifold.) If VP is a projective toric manifold, then there is a holomorphic freeaction of a complex ℓ-dimensional torus T ℓC on ZP with quotient VP .

Using the construction of a complex structure on ZK described in the previoussection, in [262] holomorphic (Seifert) fibrations with total space ZK were defined


for arbitrary complete rational simplicial fans Σ. By an application of the Borelspectral sequence to the holomorphic fibration ZK → VΣ, the Dolbeault cohomologyof ZK can be described and some Hodge numbers can be calculated explicitly.

Here we make additional assumption that the set of integral linear combinationsof the vectors a1, . . . ,am is a full-rank lattice (a discrete subgroup isomorphicto Zn) in NR

∼= Rn. We denote this lattice by NZ or simply N . This assumptionimplies that the complete simplicial fan Σ defined by the data K;a1, . . . ,am isrational. We also continue assuming that m− n is even and setting ℓ = m−n

2 .Because of our rationality assumption, the algebraic groupG is defined by (5.3).

Furthermore, since we defined N as the lattice generated by a1, . . . ,am, the groupG is isomorphic to (C×)2ℓ (i.e. there are no finite factors). We also observe that CΨlies in G as an ℓ-dimensional complex subgroup. This follows from condition (b’)of Construction 6.6.1.

The quotient construction (Section 5.4) identifies the toric variety VΣ withU(K)/G, provided that a1, . . . ,am are primitive generators of the edges of Σ.In our data K;a1, . . . ,am, the vectors a1, . . . ,am are not necessarily primitivein the lattice N generated by them. Nevertheless, the quotient U(K)/G is stillisomorphic to VΣ, see [6, Ch. II, Proposition 3.1.7]. Indeed, let a ′

i ∈ N be theprimitive generator along a i, so that ai = ria

′i for some positive integer ri. Then

we have a finite branched covering

U(K)→ U(K), (z1, . . . , zm) 7→ (zr11 , . . . , zrmm ),

which maps the group G defined by a1, . . . ,am to the group G′ defined bya ′1, . . . ,a

′m, see (5.3). We therefore obtain a covering U(K)/G → U(K)/G′ of

the toric variety VΣ ∼= U(K)/G ∼= U(K)/G′ over itself. Having this in mind, we canrelate the quotients VΣ ∼= U(K)/G and ZK

∼= U(K)/CΨ as follows:

Proposition 6.7.1. Assume that data K;a1, . . . ,am define a complete sim-plicial rational fan Σ, and let G and CΨ be the groups defined by (5.3) and (6.23).

(a) The toric variety VΣ is identified, as a topological space, with the quotientof ZK by the holomorphic action of the complex compact torus G/CΨ .

(b) If the fan Σ is regular, then VΣ is the base of a holomorphic principalbundle with total space ZK and fibre the complex compact torus G/CΨ .

Proof. To prove (a) we just observe that

VΣ = U(K)/G =(U(K)/CΨ

)/(G/CΨ ) ∼= ZK

/(G/CΨ ),

where we used Theorem 6.6.3. The quotient G/CΨ is a compact complex ℓ-torus byExample 6.6.2. To prove (b) we observe that the holomorphic action of G on U(K)is free by Proposition 5.4.6, and the same is true for the action of G/CΨ on ZK. Aholomorphic free action of the torus G/CΨ gives rise to a principal bundle.

Remark. As in the projective situation of [224], if the fan Σ is not regular,then the quotient projection ZK → VΣ of Proposition 6.7.1 (a) is a holomorphicprincipal Seifert fibration for an appropriate orbifold structure on VΣ.

Let M be a complex n-dimensional manifold. The space Ω∗C(M) of com-

plex differential forms on M decomposes into a direct sum of the subspaces of(p, q)-forms, Ω∗

C(M) =⊕

06p,q6nΩp,q(M), and there is the Dolbeault differential

∂ : Ωp,q(M)→ Ωp,q+1(M). The dimensions hp,q(M) of the Dolbeault cohomology


groups Hp,q

∂(M), 0 6 p, q 6 n, are known as the Hodge numbers of M . They are

important invariants of the complex structure of M .The Dolbeault cohomology of a compact complex ℓ-torus T ℓC is isomorphic to

an exterior algebra on 2ℓ generators:

(6.25) H∗,∗

∂(T ℓC)

∼= Λ[ξ1, . . . , ξℓ, η1, . . . , ηℓ],

where ξ1, . . . , ξℓ ∈ H1,0

∂(T ℓC) are the classes of basis holomorphic 1-forms, and

η1, . . . , ηℓ ∈ H0,1

∂(T ℓC) are the classes of basis antiholomorphic 1-forms. In par-

ticular, the Hodge numbers are given by hp,q(T ℓC) =(ℓp

)(ℓq

).

The de Rham cohomology of a toric manifold VΣ admits a Hodge decompo-sition with only nontrivial components of bidegree (p, p), 0 6 p 6 n [86, §12].This together with Theorem 5.3.1 gives the following description of the Dolbeaultcohomology:

(6.26) H∗,∗

∂(VΣ) ∼= C[v1, . . . , vm]/(IK + JΣ),

where vi ∈ H1,1

∂(VΣ) are the cohomology classes corresponding to torus-invariant

divisors (one for each one-dimensional cone of Σ), the ideal IK is generated by themonomials vi1 · · · vik for which a i1 , . . . ,a ik do not span a cone of Σ (the Stanley–Reisner ideal ofK), and JΣ is generated by the linear forms

∑mj=1〈aj ,u〉vj , u ∈ N∗.

For the Hodge numbers, hp,p(VΣ) = hp, where (h0, h1, . . . , hn) is the h-vector of K,and hp,q(VΣ) = 0 for p 6= q.

Theorem 6.7.2 ([262]). Assume that data K;a1, . . . ,am define a completerational regular fan Σ in NR

∼= Rn, m − n = 2ℓ, and let ZK be the correspondingmoment-angle manifold with a complex structure defined by Theorem 6.6.3. Thenthe Dolbeault cohomology algebra H∗,∗

∂(ZK) is isomorphic to the cohomology of the

differential bigraded algebra

(6.27)[Λ[ξ1, . . . , ξℓ, η1, . . . , ηℓ]⊗H∗,∗

∂(VΣ), d

]

with differential d of bidegree (0, 1) defined on the generators as follows:

dvi = dηj = 0, dξj = c(ξj), 1 6 i 6 m, 1 6 j 6 ℓ,

where c : H1,0

∂(T ℓC) → H2(VΣ,C) = H1,1

∂(VΣ) is the first Chern class map of the

principal T ℓC-bundle ZK → VΣ.

Proof. We use the notion of a minimal Dolbeault model of a complex man-ifold [113, §4.3]. Let [B, dB ] be such a model for VΣ, i.e. [B, dB ] is a mini-mal commutative bigraded differential algebra together with a quasi-isomorphismf : B∗,∗ → Ω∗,∗(VΣ). Consider the differential bigraded algebra

(6.28)

[Λ[ξ1, . . . , ξℓ, η1, . . . , ηℓ]⊗B, d

], where

d|B = dB, d(ξi) = c(ξi) ∈ B1,1 = H1,1

∂(VΣ), d(ηi) = 0.

By [113, Corollary 4.66], this is a model for the Dolbeault cohomology algebraof the total space ZK of the principal T ℓC-bundle ZK → VΣ, provided that VΣ isstrictly formal. Recall from [113, Definition 4.58] that a complex manifold M isstrictly formal if there exists a differential bigraded algebra [Z, δ] together with


quasi-isomorphisms

[Ω∗,∗, ∂] [Z, δ]≃oo ≃ //

≃

[Ω∗, dDR]

[H∗,∗

∂(M), 0]

linking together the de Rham algebra, the Dolbeault algebra and the Dolbeaultcohomology.

According to [260, Corollary 7.2], the toric manifold VΣ is formal in the usual(de Rham) sense. Also, the above mentioned Hodge decomposition of [86, §12]implies that VΣ satisfies the ∂∂-lemma [113, Lemma 4.24]. Therefore VΣ is strictlyformal by the same argument as [113, Theorem 4.59], and (6.28) is a model for itsDolbeault cohomology.

The usual formality of VΣ implies the existence of a quasi-isomorphismϕB : B → H∗,∗

∂(VΣ), which extends to a quasi-isomorphism

id⊗ ϕB :[Λ[ξ1, . . . , ξℓ, η1, . . . , ηℓ]⊗B, d

]→[Λ[ξ1, . . . , ξℓ, η1, . . . , ηℓ]⊗H∗,∗

∂(VΣ), d

]

by [112, Lemma 14.2]. Thus, the differential algebra[Λ[ξ1, . . . , ξℓ, η1, . . . , ηℓ] ⊗

H∗,∗

∂(VΣ), d

]provides a model for the Dolbeault cohomology of ZK, as claimed.

Remark. If VΣ is projective, then it is Kahler; in this case the model of Theo-rem 6.7.2 coincides with the model for the Dolbeault cohomology of the total spaceof a holomorphic torus principal bundle over a Kahler manifold [113, Theorem 4.65].

The first Chern class map c from Theorem 6.7.2 can be described explicitlyin terms of the map Ψ defining the complex structure on ZK. We recall the mapAC : Cm → NC, ei 7→ a i and the Gale dual (m − n) ×m-matrix Γ = (γjk) whoserows form a basis of linear relations between a1, . . . ,am. By Construction 6.6.1,ImΨ ⊂ KerAC. Denote by AnnU the annihilator of a linear subspace U ⊂ Cm,i.e. the subspace of linear functions on Cm vanishing on U .

Lemma 6.7.3. Let k be the number of zero vectors among a1, . . . ,am. The firstChern class map

c : H1,0

∂(T ℓC)→ H2(VΣ,C) = H1,1

∂(VΣ)

of the principal T ℓC-bundle ZK → VΣ is given by the composition

Ann ImΨ/AnnKerACi−−−−→ Cm/AnnKerAC

p−−−−→ Cm−k/AnnKerAC

where i is the inclusion and p is the projection forgetting the coordinates in Cm

corresponding to zero vectors.Explicitly, the map c is given on the generators of H1,0

∂(T ℓC) by

c(ξj) = µj1v1 + · · ·+ µjmvm, 1 6 j 6 ℓ,

where M = (µjk) is an ℓ×m-matrix satisfying the two conditions:

(a) ΓM t : Cℓ → C2ℓ is a monomorphism;(b) MΨ = 0.

Proof. Let A∗C : N

∗C → Cm, u 7→ (〈a1,u〉, . . . , 〈am,u〉), be the dual map. We

have H1(T ℓC;C) = (KerAC)∗ = Cm/ ImA∗

C and H2(VΣ;C) = Cm−k/ ImA∗C. The

first Chern class map c : H1(T ℓC;C) → H2(VΣ;C) (the transgression) is then givenby p : Cm/ ImA∗

C → Cm−k/ ImA∗C. In order to separate the holomorphic part of c


we need to identify the subspace of holomorphic differentials H1,0

∂(T ℓC)

∼= Cℓ inside

the space of all 1-forms H1(T ℓC;C)∼= C2ℓ. Since

T ℓC = G/CΨ = (Ker expAC)/(exp ImΨ),

holomorphic differentials on T ℓC correspond to C-linear functions on KerAC whichvanish on ImΨ . The space of functions on KerAC is Cm/ ImA∗

C = Cm/AnnKerAC,and the functions vanishing on ImΨ form the subspace Ann ImΨ/AnnKerAC.Condition (b) says exactly that the linear functions on Cm corresponding to therows of M vanish on ImΨ . Condition (a) says that the rows of M constitute a basisin the complement of AnnKerAC in Ann ImΨ .

It is interesting to compare Theorem 6.7.2 with the following description of thede Rham cohomology of ZK:

Theorem 6.7.4. Let ZK and VΣ be as in Theorem 6.7.2. The de Rham coho-mology H∗(ZK) is isomorphic to the cohomology of the differential graded algebra

[Λ[u1, . . . , um−n]⊗H∗(VΣ), d

],

with deg uj = 1, deg vi = 2, and differential d defined on the generators as

dvi = 0, duj = γj1v1 + · · ·+ γjmvm, 1 6 j 6 m− n.

Proof. The de Rham cohomology of the manifold ZK is isomorphic to itscellular cohomology (with coefficients in R). By Theorem 4.5.4,

H∗(ZK) ∼= TorR[v1,...,vm]

(R[K],R

)

Since Σ is a complete fan, K = KΣ is a sphere triangulation, and therefore the facering R[K] = R[v1, . . . , vm]/IK is Cohen–Macaulay by Corollary 3.3.13. The idealJΣ is generated by a regular sequence, so we obtain by Lemma A.3.5,

TorR[v1,...,vm]

(R[K],R

) ∼= TorR[v1,...,vm]/JΣ

(R[K]/JΣ,R

).

Since R[v1, . . . , vm]/JΣ is a polynomial ring in m − n variables, Lemma A.2.10implies that the Tor-algebra above is isomorphic to

H[Λ[u1, . . . , um−n]⊗ R[K]/JΣ, d

] ∼= H[Λ[u1, . . . , um−n]⊗H∗(VΣ), d

]

where the explicit form of the differential d follows from the definition of the Galedual configuration Γ = (γ1, . . . , γm).

There are two classical spectral sequences for the Dolbeault cohomology. First,the Borel spectral sequence [34] of a holomorphic bundle E → B with a compactKahler fibre F , which has E2 = H∂(B)⊗H∂(F ) and converges to H∂(E). Second,the Frolicher spectral sequence [135, §3.5], whose E1-term is the Dolbeault coho-mology of a complex manifold M and which converges to the de Rham cohomologyof M . Theorem 6.7.2 implies a collapse result for these spectral sequences:

Corollary 6.7.5.

(a) The Borel spectral sequence of the holomorphic principal bundle ZK → VΣcollapses at the E3-term, i.e. E3 = E∞;

(b) the Frolicher spectral sequence of ZK collapses at the E2-term.


Proof. To prove (a) we just observe that the differential algebra (6.27) is theE2-term of the Borel spectral sequence, and its cohomology is the E3-term.

By comparing the Dolbeault and de Rham cohomology algebras of ZK givenby Theorems 6.7.2 and 6.7.4 we observe that the elements η1, . . . , ηℓ ∈ E0,1

1 cannotsurvive in the E∞-term of the Frolicher spectral sequence. The only possible non-trivial differential on them is d1 : E

0,11 → E1,1

1 . By Theorem 6.7.4, the cohomologyalgebra of [E1, d1] is exactly the de Rham cohomology of ZK, proving (b).

Theorem 6.7.4 can also be interpreted as a collapse result for the Leray–Serrespectral sequence of the principal Tm−n-bundle ZK → VΣ.

In order to proceed with calculation of Hodge numbers, we need the followingbounds for the dimension of Ker c in Lemma 6.7.3:

Lemma 6.7.6. Let k be the number of zero vectors among a1, . . . ,am. Then

k − ℓ 6 dimC Ker(c : H1,0

∂(T ℓC)→ H1,1

∂(VΣ)

)6 k

2 .

In particular, if k 6 1 then c is monomorphism.

Proof. Consider the commutative diagram

Ann ImΨ/AnnKerACi−−−−→ Cm/AnnKerAC

p−−−−→ Cm−k/AnnKerACy∼=

yRe

yRe

Rm−n Rm−n p′−−−−→ Rm−n−k.

The composition Re ·i is an R-linear isomorphism, as it has the form H1,0

∂(T ℓC) →

H1(T ℓC,C)→ H1(T ℓC,R), and any real-valued function on the lattice Γ defining thetorus T ℓC = Cℓ/Γ is the real part of the restriction to Γ of a C-linear function on Cℓ.

Since the diagram above is commutative, the kernel of c = p i has real dimen-sion at most k, which implies the upper bound on its complex dimension. For thelower bound, dimC Ker c > dimH1,0

∂(T ℓC)−dimH1,1

∂(VΣ) = ℓ− (2ℓ−k) = k− ℓ.

Theorem 6.7.7. Let ZK be as in Theorem 6.7.2, and let k be the number ofzero vectors among a1, . . . ,am. Then the Hodge numbers hp,q = hp,q(ZK) satisfy

(a)(k−ℓp

)6 hp,0 6

([k/2]p

)for p > 0; in particular, hp,0 = 0 for p > 0 if k 6 1;

(b) h0,q =(ℓq

)for q > 0;

(c) h1,q = (ℓ− k)(ℓ

q−1

)+ h1,0

(ℓ+1q

)for q > 1;

(d) ℓ(3ℓ+1)2 − h2(K) − ℓk + (ℓ+ 1)h2,0 6 h2,1 6

ℓ(3ℓ+1)2 − ℓk + (ℓ+ 1)h2,0.

Proof. Let Ap,q denote the bidegree (p, q) component of the differential al-gebra from Theorem 6.7.2, and let Zp,q ⊂ Ap,q denote the subspace of cocycles.Then d1,0 : A1,0 → Z1,1 coincides with the map c, and the required bounds forh1,0 = Kerd1,0 are already established in Lemma 6.7.6. Since hp,0 = dimKer dp,0,and Ker dp,0 is the pth exterior power of the space Ker d1,0, statement (a) follows.

The differential is trivial on A0,q, hence h0,q = dimA0,q, proving (b).The space Z1,1 is spanned by the cocycles vi and ξiηj with ξi ∈ Ker d1,0. Hence

dimZ1,1 = 2ℓ − k + h1,0ℓ. Also, dim d(A1,0) = ℓ − h1,0, therefore, h1,1 = ℓ − k +

h1,0(ℓ+1). Similarly, dimZ1,q = (2ℓ−k)(ℓ

q−1

)+h1,0

(ℓq

)(with basis of viηj1 · · · ηjq−1

and ξiηj1 · · · ηjq where ξi ∈ Ker d1,0, j1 < · · · < jq), and d : A1,q−1 → Z1,q hits a

subspace of dimension (ℓ − h1,0)(ℓ

q−1

). This proves (c).


We have A2,1 = U ⊕W , where U has basis of monomials ξivj and W has basisof monomials ξiξjηk. Therefore,

(6.29) h2,1 = dimU − dim dU + dimW − dim dW − dim dA2,0.

Now dimU = ℓ(2ℓ − k), 0 6 dim dU 6 h2(K) (since dU ⊂ H2,2

∂(VΣ)), dimW −

dim dW = dimKer d|W = ℓh2,0, and dim dA2,0 =(ℓ2

)− h2,0. By substituting all

this into (6.29) we obtain the inequalities of (d).

Remark. At most one ghost vertex needs to be added to K to make dimZK =m + n even. Since hp,0(ZK) = 0 when k 6 1, the manifold ZK does not haveholomorphic forms of any degree in this case.

If ZK is a torus (so that K is empty), then m = k = 2ℓ, and h1,0(ZK) =h0,1(ZK) = ℓ. Otherwise Theorem 6.7.7 implies that h1,0(ZK) < h0,1(ZK), andtherefore ZK is not Kahler.

Example 6.7.8. Let ZK∼= S1 × S2n+1 be a Hopf manifold of Example 6.6.4.

Our rationality assumption is that a1 . . . ,an+2 span an n-dimensional lattice Nin NR

∼= Rn; in particular, the fan Σ defined by the proper subsets of a1, . . . ,an+1

is rational. We assume further that Σ is regular (this is equivalent to the conditiona1+ · · ·+an+1 = 0), so that Σ is a the normal fan of a Delzant n-dimensional sim-plex ∆n. We have VΣ = CPn, and (6.26) describes its cohomology as the quotientof C[v1, . . . , vn+2] by the two ideals: I generated by v1 · · · vn+1 and vn+2, and Jgenerated by v1 − vn+1, . . . , vn − vn+1. The differential algebra of Theorem 6.7.2is therefore given by

[Λ[ξ, η] ⊗ C[t]/tn+1, d

], with dt = dη = 0 and dξ = t for a

proper choice of t. The nontrivial cohomology classes are represented by the cocy-cles 1, η, ξtn and ξηtn, which gives the following nonzero Hodge numbers of ZK:h0,0 = h0,1 = hn+1,n = hn+1,n+1 = 1. Observe that the Dolbeault cohomology andHodge numbers do not depend on a choice of complex structure (the map Ψ).

Example 6.7.9 (Calabi–Eckmann manifold). Let K;a1, . . . ,an+2 be thedata defining the normal fan of the product P = ∆p×∆q of two Delzant simpliceswith p+ q = n, 1 6 p 6 q 6 n− 1. That is, a1, . . . ,ap,ap+2, . . . ,an+1 is a basis oflattice N and there are two relations a1+ · · ·+ap+1 = 0 and ap+2+ · · ·+an+2 = 0.The corresponding toric variety VΣ is CP p × CP q and its cohomology ring is iso-morphic to C[x, y]/(xp+1, yq+1). Consider the map

Ψ : C→ Cn+2, w 7→ (w, . . . , w, αw, . . . , αw),

where α ∈ C\R and αw appears q+1 times. The map Ψ satisfies the conditions ofConstruction 6.6.1. The resulting complex structure on ZP ∼= S2p+1×S2q+1 is thatof a Calabi–Eckmann manifold. We denote complex manifolds obtained in this wayby CE (p, q) (the complex structure depends on the choice of Ψ , but we do not reflectthis in the notation). Each manifold CE (p, q) is the total space of a holomorphicprincipal bundle over CP p × CP q with fibre the complex 1-torus C/(Z⊕ αZ).

Theorem 6.7.2 and Lemma 6.7.3 provide the following description of the Dol-beault cohomology of CE (p, q):

H∗,∗

∂

(CE (p, q)

) ∼= H[Λ[ξ, η]⊗ C[x, y]/(xp+1, yq+1), d

],

where dx = dy = dη = 0 and dξ = x − y for an appropriate choice of x, y. Wetherefore obtain

(6.30) H∗,∗

∂

(CE (p, q)

) ∼= Λ[ω, η]⊗ C[x]/(xp+1),


where ω ∈ Hq+1,q

∂

(CE (p, q)

)is the cohomology class of the cocycle ξ x

q+1−yq+1

x−y . This

calculation is originally due to [34, §9]. We note that the Dolbeault cohomologyof a Calabi–Eckmann manifold depends only on p, q and does not depend on thecomplex parameter α (or the map Ψ).

Example 6.7.10. Now let P = ∆1 × ∆1 × ∆2 × ∆2. Then the moment-angle manifold ZP has two structures of a product of Calabi–Eckmann manifolds,namely, CE (1, 1)×CE (2, 2) and CE (1, 2)× CE (1, 2). Using isomorphism (6.30) weobserve that these two complex manifolds have different Hodge numbers: h2,1 = 1in the first case and h2,1 = 0 in the second. This shows that the choice of themap Ψ affects not only the complex structure of ZK, but also its Hodge numbers,unlike the previous examples of complex tori, Hopf and Calabi–Eckmann manifolds.Certainly it is not highly surprising from the complex-analytic point of view.

6.8. Hamiltonian-minimal Lagrangian submanifolds

In this last section we apply the accumulated knowledge on topology of moment-angle manifolds in a somewhat different area, Lagrangian geometry. Systems of realquadrics, which we used in Sections 6.1 and 6.2 to define moment-angle manifolds,also give rise to a family of Hamiltonian-minimal Lagrangian submanifolds in acomplex space or more general toric varieties.

Hamiltonian minimality (H-minimality for short) for Lagrangian submanifoldsis a symplectic analogue of minimality in Riemannian geometry. A Lagrangianimmersion is called H-minimal if the variations of its volume along all Hamiltonianvector fields are zero. This notion was introduced in the work of Y.-G. Oh [251]in connection with the celebrated Arnold conjecture on the number of fixed pointsof a Hamiltonian symplectomorphism. The simplest example of an H-minimalLagrangian submanifold is the coordinate torus [251] S1

r1 × · · · ×S1rm ⊂ Cm, where

S1rk

denotes the circle of radius rk > 0 in the kth coordinate subspace of Cm.More examples of H-minimal Lagrangian submanifolds in a complex space wereconstructed in the works [68], [156], [5], among others.

In [229] Mironov suggested a general construction of H-minimal Lagrangianimmersions N # Cm from intersections of real quadrics. These systems of quadricsare the same as those we used to define moment-angle manifolds, and therefore onecan apply toric methods for analysing the topological structure of N . In [230] aneffective criterion was obtained for N # Cm to be an embedding: the polytopecorresponding to the intersection of quadrics must be Delzant. As a consequence,any Delzant polytope gives rise to an H-minimal Lagrangian submanifold N ⊂ Cm.As in the case of moment-angle manifolds, the topology of N is quite complicatedeven for low-dimensional polytopes: for example, a Delzant 5-gon gives rise to amanifold N which is the total space of a bundle over a 3-torus with fibre a surfaceof genus 5. Furthermore, by combining Mironov’s construction with symplecticreduction, a new family of H-minimal Lagrangian submanifolds in of toric varietieswas defined in [231]. This family includes many previously constructed explicitexamples in Cm and CPm−1.

Preliminaries. Let (M,ω) be a symplectic manifold of dimension 2n. Animmersion i : N # M of an n-dimensional manifold N is called Lagrangian ifi∗(ω) = 0. If i is an embedding, then i(N) is a Lagrangian submanifold of M . Avector field X on M is Hamiltonian if the 1-form ω(X, · ) is exact.

6.8. HAMILTONIAN-MINIMAL LAGRANGIAN SUBMANIFOLDS 237

Now assume thatM is Kahler, so that it has compatible Riemannian metric andsymplectic structure. A Lagrangian immersion i : N # M is called Hamiltonianminimal (H-minimal) if the variations of the volume of i(N) along all Hamiltonianvector fields with compact support are zero, that is,

d

dtvol(it(N)

)∣∣t=0

= 0,

where it(N) is a deformation of i(N) along a Hamiltonian vector field, i0(N) =i(N), and vol(it(N)) is the volume of the deformed part of it(N). An immersion iis minimal if the variations of the volume of i(N) along all vector fields are zero.

Our basic example is M = Cm with the Hermitian metric 2∑mk=1 dzk ⊗ dzk.

Its imaginary part is the symplectic form of Example 5.5.1. In the end we considera more general case when M is a toric manifold.

The construction. We consider an intersection of quadrics similar to (6.5),but in the real space:

(6.31) R =u = (u1, . . . , um) ∈ Rm :

m∑

k=1

γjku2k = δj , for 1 6 j 6 m− n

.

We assume the nondegeneracy and rationality conditions on the coefficientvectors γi = (γ1i, . . . , γm−n,i)

t ∈ Rm−n, i = 1, . . . ,m:

(a) δ ∈ R>〈γ1, . . . , γm〉;(b) if δ ∈ R>〈γi1 , . . . γik〉, then k > m− n;(c) the vectors γ1, . . . , γm generate a lattice L ∼= Zm−n in Rm−n.

These conditions guarantee that R is a smooth n-dimensional submanifold inRm (by the argument of Proposition 6.1.4) and that

TΓ =(e2πi〈γ1,ϕ〉, . . . , e2πi〈γm,ϕ〉

)∈ Tm

is an (m − n)-dimensional torus subgroup in Tm. We identify the torus TΓ withRm−n/L∗ and represent its elements by ϕ ∈ Rm−n. We also define

DΓ =1

2L∗/L∗ ∼= (Z2)

m−n.

Note that DΓ embeds canonically as a subgroup in TΓ .Now we view the intersection R as a subset in the intersection Z or Hermitian

quadrics given by (6.5), or as a subset in the whole space Cm. Then we ‘spread’R by the action of TΓ , that is, consider the set of TΓ -orbits through R. Moreprecisely, we consider the map

j : R× TΓ −→ Cm,

(u , ϕ) 7→ u · ϕ =(u1e

2πi〈γ1,ϕ〉, . . . , ume2πi〈γm,ϕ〉

)

and observe that j(R×TΓ ) ⊂ Z. We let DΓ act on RΓ ×TΓ diagonally; this actionis free, since it is free on the second factor. The quotient

N = R×DΓTΓ

is an m-dimensional manifold.For any u = (u1, . . . , um) ∈ R, we have the sublattice

Lu = Z〈γk : uk 6= 0〉 ⊂ L = Z〈γ1, . . . , γm〉.The set of TΓ -orbits through R is an immersion of N :


Lemma 6.8.1.

(a) The map j : R× TΓ → Cm induces an immersion i : N # Cm.(b) The immersion i is an embedding if and only if Lu = L for any u ∈ R.

Proof. Take u ∈ R, ϕ ∈ TΓ and g ∈ DΓ . We have u ·g ∈ R, and j(u ·g, gϕ) =u ·g2ϕ = u ·ϕ = j(u , ϕ). Hence, the map j is constant on DΓ -orbits, and thereforeinduces a map of the quotient N = (R× TΓ )/DΓ , which we denote by i.

Assume that j(u , ϕ) = j(u ′, ϕ′). Then Lu = Lu ′ and

(6.32) uke2πi〈γk,ϕ〉 = u′ke

2πi〈γk,ϕ′〉 for k = 1, . . . ,m.

Since both uk and u′k are real, this implies that e2πi〈γk,ϕ−ϕ′〉 = ±1 whenever uk 6= 0,

or, equivalently, ϕ − ϕ′ ∈ 12L

∗u/L

∗. In other words, (6.32) implies that u ′ = u · gand ϕ′ = gϕ for some g ∈ 1

2L∗u/L

∗. The latter is a finite group by Lemma 6.3.2;hence the preimage of any point of Cm under j consists of a finite number of points.If Lu = L, then 1

2L∗u/L

∗ = 12L

∗/L∗ = DΓ ; hence (u , ϕ) and (u ′, ϕ′) represent thesame point in N . Statement (b) follows; to prove (a), it remains to observe thatwe have Lu = L for generic u (with all coordinates nonzero).

Theorem 6.8.2 ([229, Theorem 1]). The immersion i : N # Cm is H-minimalLagrangian. Moreover, if

∑mk=1 γk = 0, then i is a minimal Lagrangian immersion.

Proof. We only prove that i is a Lagrangian immersion here. Let

(x , ϕ) 7→ z (x , ϕ) =(u1(x )e

2πi〈γ1,ϕ〉, . . . , um(x )e2πi〈γm,ϕ〉

)

be a local coordinate system on N = R×DΓTΓ , where x = (x1, . . . , xn) ∈ Rn and

ϕ = (ϕ1, . . . , ϕm−n) ∈ Rm−n. Let 〈ξ, η〉C =∑m

i=1 ξiηi = 〈ξ, η〉 + iω(ξ, η) be theHermitian scalar product of ξ, η ∈ Cm. Then

⟨ ∂z∂xk

,∂z

∂ϕj

⟩C= 2πi

(γj1u1

∂u1∂xk

+ · · ·+ γjmum∂um∂xk

)= 0

where the second identity follows by differentiating the quadrics equations (6.31).Also,

⟨∂z∂xk

, ∂z∂xj

⟩C∈ R and

⟨∂z∂ϕk

, ∂z∂ϕj

⟩C∈ R. It follows that

ω( ∂z∂xk

,∂z

∂ϕj

)= ω

( ∂z∂xk

,∂z

∂xj

)= ω

( ∂z∂ϕk

,∂z

∂ϕj

)= 0,

i.e. the restriction of the symplectic form to the tangent space of N is zero.

Remark. The identity∑mk=1 γk = 0 can not hold for a compact R (or N).

We recall from Theorem 6.1.5 that a nonsingular intersection of quadrics (6.5)or (6.31) defines a simple polyhedron (1.1), and Z is identified with the moment-angle manifold ZP . Now we can summarise the results of the previous sections inthe following criterion for i : N → Cm to be an embedding:

Theorem 6.8.3. Let Z and R be the intersections of Hermitian and realquadrics defined by (6.5) and (6.31), respectively, satisfying conditions (a)– (c)above. Let P be the associated simple polyhedron, and N = R ×DΓ

TΓ . Thefollowing conditions are equivalent:

(a) i : N → Cm is an embedding of an H-minimal Lagrangian submanifold;(b) Lu = L for any u ∈ R;(c) the torus TΓ acts freely on the moment-angle manifold Z = ZP ;


(d) P is a Delzant polyhedron.

Proof. Equivalence (a)⇔ (b) follows from Lemma 6.8.1 and Theorem 6.8.2.Equivalence (b)⇔ (c) is Lemma 6.3.2, and (c)⇔ (d) is Theorem 6.3.1 (c).

Topology of Lagrangian submanifolds N ⊂ Cm. We start by reviewingthree simple properties linking the topological structure of N to that of the inter-sections of quadrics Z and R.

Proposition 6.8.4.

(a) The immersion of N in Cm factors as N # Z → Cm;(b) N is the total space of a bundle over the torus Tm−n with fibre R;(c) if N → Cm is an embedding, then N is the total space of a principal

Tm−n-bundle over the n-dimensional manifold R/DΓ .

Proof. Statement (a) is clear. Since DΓ acts freely on TΓ , the projectionN = R ×DΓ

TΓ → TΓ /DΓ onto the second factor is a fibre bundle with fibre R.Then (b) follows from the fact that TΓ /DΓ

∼= Tm−n.If N → Cm is an embedding, then TΓ acts freely on Z by Theorem 6.8.3 and the

action of DΓ on R is also free. Therefore, the projection N = R×DΓTΓ →R/DΓ

onto the first factor is a principal TΓ -bundle, which proves (c).

Remark. The quotient R/DΓ is a real toric variety, or a small cover, over thecorresponding polytope P , see [90] and [55].

Example 6.8.5 (one quadric). Suppose that R is given by a single equation

(6.33) γ1u21 + · · ·+ γmu

2m = δ

in Rm. We assume thatR is compact, so that γi and δ are positive reals,R ∼= Sm−1,and the corresponding polytope P is an n-simplex ∆n. Then N ∼= Sm−1 ×Z2 S

1,where the generator of Z2 acts by the standard free involution on S1 and by acertain involution τ on Sm−1. The topological type of N depends on τ . Namely,

N ∼=Sm−1 × S1 if τ preserves the orientation of Sm−1,

Km if τ reverses the orientation of Sm−1,

where Km is known as the m-dimensional Klein bottle.

Proposition 6.8.6. Let m − n = 1 (one quadric). We obtain an H-minimalLagrangian embedding of N ∼= Sm−1 ×Z2 S

1 in Cm if and only if γ1 = · · · = γmin (6.33). In this case, the topological type of N = N(m) depends only on the parityof m and is given by

N(m) ∼= Sm−1 × S1 if m is even,

N(m) ∼= Km if m is odd.

Proof. Since there exists u ∈ R with only one nonzero coordinate, Theo-rem 6.8.3 implies that N embeds in Cm if only if γi generates the same lattice asthe whole set γ1, . . . , γm for each i. Therefore, γ1 = · · · = γm. In this case DΓ

∼= Z2

acts by the standard antipodal involution on Sm−1, which preserves orientation ifm is even and reverses orientation otherwise.

Both examples ofH-minimal Lagrangian embeddings given by Proposition 6.8.6are well known. The Klein bottle Km with even m does not admit Lagrangianembeddings in Cm (see [242] and [286]).


Example 6.8.7 (two quadrics). In the case m − n = 2, the topology of Rand N can be described completely by analysing the action of the two commutinginvolutions on the intersection of quadrics. We consider the compact case here.

Using Proposition 1.2.7, we write R in the form

(6.34)γ11u

21 + · · ·+ γ1mu

2m = c,

γ21u21 + · · ·+ γ2mu

2m = 0,

where c > 0 and γ1i > 0 for all i.

Proposition 6.8.8. There is a number p, 0 < p < m, such that γ2i > 0 fori = 1, . . . , p and γ2i < 0 for i = p + 1, . . . ,m in (6.34), possibly after reorderingthe coordinates u1, . . . , um. The corresponding manifold R = R(p, q), where q =m− p, is diffeomorphic to Sp−1 × Sq−1. Its associated polytope P either coincideswith ∆m−2 (if one of the inequalities in (1.1) is redundant) or is combinatoriallyequivalent to the product ∆p−1 ×∆q−1 (if there are no redundant inequalities).

Proof. We observe that γ2i 6= 0 for all i in (6.34), as γ2i = 0 implies that

the vector δ =( c0

)is in the cone generated by one γi, which contradicts Proposi-

tion 6.1.4 (b). By reordering the coordinates, we can achieve that the first p of γ2iare positive and the rest are negative. Then 1 < p < m, because otherwise (6.34)is empty. Now, (6.34) is the intersection of the cone over the product of two ellip-soids of dimensions p− 1 and q− 1 (given by the second quadric) with an (m− 1)-dimensional ellipsoid (given by the first quadric). Therefore,R(p, q) ∼= Sp−1×Sp−1.The statement about the polytope follows from the combinatorial fact that a simplen-polytope with up to n + 2 facets is combinatorially equivalent to a product ofsimplices; the case of one redundant inequality corresponds to p = 1 or q = 1.

An element ϕ ∈ DΓ = 12L

∗/L∗ ∼= Z2 × Z2 acts on R(p, q) by

(u1, . . . , um) 7→ (ε1(ϕ)u1, . . . , εm(ϕ)um),

where εk(ϕ) = e2πi〈γk,ϕ〉 = ±1 for 1 6 k 6 m.

Lemma 6.8.9. Suppose that DΓ acts on R(p, q) freely and εi(ϕ) = 1 for some i,1 6 i 6 p, and ϕ ∈ DΓ . Then εl(ϕ) = −1 for all l with p+ 1 6 l 6 m.

Proof. Assume the opposite, that is, εi(ϕ) = 1 for some 1 6 i 6 p andεj(ϕ) = 1 for some p + 1 6 j 6 m. Then γ2i > 0 and γ2j < 0 in (6.34), so we canchoose u ∈ R(p, q) whose only nonzero coordinates are ui and uj . The elementϕ ∈ DΓ fixes this u , leading to contradiction.

Lemma 6.8.10. Suppose DΓ acts on R(p, q) freely. Then there exist two gen-erating involutions ϕ1, ϕ2 ∈ DΓ

∼= Z2 × Z2 whose action on R(p, q) is described byeither (a) or (b) below, possibly after reordering the coordinates:

(a)ϕ1 : (u1, . . . , um) 7→ (u1, . . . , uk,−uk+1, . . . ,−up,−up+1, . . . ,−um),ϕ2 : (u1, . . . , um) 7→ (−u1, . . . ,−uk, uk+1, . . . , up,−up+1, . . . ,−um);

(b)ϕ1 : (u1, . . . , um) 7→ (−u1, . . . ,−up, up+1, . . . , up+l,−up+l+1, . . . ,−um),ϕ2 : (u1, . . . , um) 7→ (−u1, . . . ,−up,−up+1, . . . ,−up+l, up+l+1, . . . , um);

here 0 6 k 6 p and 0 6 l 6 q.


Proof. By Lemma 6.8.9, for each of the three nonzero elements ϕ ∈ DΓ ,we have either εi(ϕ) = −1 for 1 6 i 6 p or εi(ϕ) = −1 for p + 1 6 i 6 m.Therefore, we may choose two different nonzero elements ϕ1, ϕ2 ∈ DΓ such thateither εi(ϕj) = −1 for j = 1, 2 and p + 1 6 i 6 m, or εi(ϕj) = −1 for j = 1, 2and 1 6 i 6 p. This corresponds to the cases (a) and (b) above, respectively. Inthe former case, after reordering the coordinates, we may assume that ϕ1 acts asin (a). Then ϕ2 also acts as in (a), since otherwise the composition ϕ1 · ϕ2 cannotact freely by Lemma 6.8.9. The second case is treated similarly.

Each of the actions of DΓ described in Lemma 6.8.10 can be realised by aparticular intersection of quadrics (6.34). For example, the system of quadrics

(6.35)2u21 + · · ·+ 2u2k + u2k+1 + · · ·+ u2p + u2p+1 + · · ·+ u2m = 3,

u21 + · · ·+ u2k + 2u2k+1 + · · ·+ 2u2p − u2p+1 − · · · − u2m = 0

gives the first action of Lemma 6.8.10; the second action is realised similarly. Notethat the lattice L corresponding to (6.35) is a sublattice of index 3 in Z2. We canrewrite (6.35) as

(6.36)u21 + · · ·+ u2k + u2k+1 + · · ·+ u2p = 1,

u21 + · · ·+ u2k +u2p+1 + · · ·+ u2m = 2,

in which case L = Z2. The action of the two involutions ψ1, ψ2 ∈ DΓ = 12Z

2/Z2

corresponding to the standard basis vectors of 12Z

2 is given by

(6.37)ψ1 : (u1, . . . , um) 7→ (−u1, . . . ,−uk,−uk+1, . . . ,−up, up+1, . . . , um),

ψ2 : (u1, . . . , um) 7→ (−u1, . . . ,−uk, uk+1, . . . , up,−up+1, . . . ,−um).We denote the manifold N corresponding to (6.36) by Nk(p, q). We have

(6.38) Nk(p, q) ∼= (Sp−1 × Sq−1)×Z2×Z2 (S1 × S1),

where the action of the two involutions on Sp−1 × Sq−1 is given by ψ1, ψ2 above.Note that ψ1 acts trivially on Sq−1 and acts antipodally on Sp−1. Therefore,

Nk(p, q) ∼= N(p)×Z2 (Sq−1 × S1),

where N(p) is the manifold from Proposition 6.8.6. If k = 0 then the secondinvolution ψ2 acts trivially on N(p), and N0(p, q) coincides with the product N(p)×N(q) of the two manifolds from Example 6.8.5. In general, the projection

Nk(p, q)→ Sq−1 ×Z2 S1 = N(q)

describes Nk(p, q) as the total space of a fibration over N(q) with fibre N(p).We summarise the above facts and observations in the following topological

classification result for compact H-minimal Lagrangian submanifolds N ⊂ Cm

obtained from intersections of two quadrics.

Theorem 6.8.11. Let N → Cm be the embedding of the H-minimal Lagrangiansubmanifold corresponding to a compact intersection of two quadrics. Then N isdiffeomorphic to some Nk(p, q) given by (6.38), where p+ q = m, 0 < p < m and0 6 k 6 p. Moreover, every such triple (k, p, q) can be realised by N .

In the case of up to two quadrics considered above, the topology of R is rela-tively simple, and in order to analyse the topology of N , one only needs to describethe action of involutions on R. When the number of quadrics is more than two,the topology of R becomes an issue as well.


Example 6.8.12 (three quadrics). In the case m − n = 3, the topology ofcompact manifolds R and Z was fully described in [197, Theorem 2]. Each ofthese manifolds is diffeomorphic to a product of three spheres or to a connectedsum of products of spheres with two spheres in each product.

Note that, for m − n = 3, the manifolds R (or Z) can be distinguished topo-logically by looking at the planar Gale diagrams of the corresponding simple poly-topes P (see Section 1.2). This chimes with the classification of simple n-polytopeswith n+ 3 facets, well-known in combinatorial geometry [325, §6.5].

The smallest polytope with m − n = 3 is a pentagon. It has many Delzantrealisations, for instance,

P =(x1, x2) ∈ R2 : x1 > 0, x2 > 0, −x1+2 > 0, −x2+2 > 0, −x1−x2+3 > 0

.

In this case, R is an oriented surface of genus 5 (see Proposition 4.1.8), and themoment-angle manifold Z is diffeomorphic to a connected sum of 5 copies of S3×S4.

We therefore obtain an H-minimal Lagrangian submanifold N ⊂ C5, which isthe total space of a bundle over T 3 with fibre a surface of genus 5.

Now assume that the polytope P associated with intersection of quadrics (6.31)is a polygon (i.e., n = 2). If there are no redundant inequalities, then P is an m-gonandR is an orientable surface Sg of genus g = 1+2m−3(m−4) by Proposition 4.1.8.If there are k redundant inequalities, then P is an (m − k)-gon. In this case R ∼=R′×(S0)k, where R′ corresponds to an (m−k)-gon without redundant inequalities.That is, R is a disjoint union of 2k surfaces of genus 1 + 2m−k−3(m− k − 4).

The corresponding H-minimal submanifold N ⊂ Cm is the total space of abundle over Tm−2 with fibre Sg. This is an aspherical manifold for m > 4.

Generalisation to toric manifolds. Consider two sets of quadrics:

ZΓ =z ∈ Cm :

∑m

k=1γk|zk|2 = c

, γk, c ∈ Rm−n;

Z∆ =z ∈ Cm :

∑m

k=1δk|zk|2 = d

, δk,d ∈ Rm−ℓ;

such that ZΓ , Z∆ and ZΓ ∩ Z∆ satisfy the nondegeneracy and rationality condi-tions (a)–(c) from the beginning of this section. Assume also that the polyhedracorresponding to ZΓ , Z∆ and ZΓ ∩ Z∆ are Delzant.

The idea is to use the first set of quadrics to produce a toric manifold V viasymplectic reduction (as described in Section 5.5), and then use the second set ofquadrics to define an H-minimal Lagrangian submanifold in V .

Construction 6.8.13. Define the real intersections of quadrics RΓ , R∆, thetori TΓ ∼= Tm−n, T∆ ∼= Tm−ℓ, and the groups DΓ

∼= Zm−n2 , D∆

∼= Zm−ℓ2 as before.

We consider the toric variety V obtained as the symplectic quotient of Cm bythe torus corresponding to the first set of quadrics: V = ZΓ /TΓ . It is a Kahlermanifold of real dimension 2n. The quotient RΓ /DΓ is the set of real points of V(the fixed point set of the complex conjugation, or the real toric manifold); it hasdimension n. Consider the subset of RΓ /DΓ defined by the second set of quadrics:

S = (RΓ ∩R∆)/DΓ ,

we have dimS = n+ ℓ−m. Finally, define the n-dimensional submanifold of V :

N = S ×D∆T∆.


Theorem 6.8.14. N is an H-minimal Lagrangian submanifold in the toricmanifold V .

Proof. Let V be the symplectic quotient of V by the torus corresponding to

the second set of quadrics, that is, V = (V ∩ Z∆)/T∆ = (ZΓ ∩ Z∆)/(TΓ × T∆). Itis a toric manifold of real dimension 2(n+ ℓ−m). The submanifold of real points

N = N/T∆ = (RΓ ∩R∆)/(DΓ ×D∆) → (ZΓ ∩ Z∆)/(TΓ × T∆) = V

is the fixed point set of the complex conjugation, hence it is a totally geodesic

submanifold. In particular, N is a minimal submanifold in V . According to [99,Corollary 2.7], N is an H-minimal submanifold in V .

Example 6.8.15.1. If m − ℓ = 0, i.e. Z∆ = ∅, then V = Cm and we obtain the original

construction of H-minimal Lagrangian submanifolds N in Cm.2. If m − n = 0, i.e. ZΓ = ∅, then N is set of real points of V . It is minimal

(totally geodesic).3. If m−ℓ = 1, i.e. Z∆ ∼= S2m−1, then we get H-minimal Lagrangian submani-

folds in V = CPm−1. This includes the families of projective examples constructedin [228], [201] and [232].

CHAPTER 7

Half-dimensional torus actions

In this chapter we study several topological generalisations of toric varieties.All of them are smooth manifolds with an action of a compact torus, the so-calledT -manifolds. Most of the results here concern the case when the dimension of theacting torus is half the dimension of the manifold.

A compact nonsingular toric variety (or toric manifold) VΣ of real dimension 2nhas an action of n-dimensional torus T n obtained by restricting the action of thealgebraic torus (C×)n. The action of T n on VΣ is locally standard, that is, it locallylooks like the standard coordinatewise action of T n on Cn (more precise definitionsare given below). If the variety VΣ is projective, then the quotient VΣ/T

n can beidentified with a simple convex n-polytope P via the moment map. In general,the quotient VΣ/T

n may be not a polytope, even combinatorially, but it still has aface structure of a manifold with corners, which is dual to the face structure of thesimplicial fan Σ. In particular, the fixed points of the T n-action on VΣ correspondto the vertices of the quotient VΣ/T

n. These basic properties of the torus actioncan be taken as the starting point for several different topological generalisationsof toric manifolds.

A quasitoric manifold is a 2n-dimensional manifold M with a locally standardaction of T n such that the quotient M/T n can be identified with a simple n-polytope P . This class of manifolds was introduced in the seminal paper [90] ofDavis and Januszkiewicz. They showed, among other things, that the cohomologyring structure of a quasitoric manifold is exactly the same as that of a toric manifold(see Theorem 5.3.1). Quasitoric manifolds have been studied intensively since thesecond half of 1990s, and the work [53] summarised these early developments andemphasised the role of moment-angle manifolds ZP .

Around the same time, an alternative way to generalise toric varieties wasdeveloped in the works of Masuda [206] and Hattori–Masuda [153], which ledto a wider class of torus manifolds. Along with the usual conditions on the T n-action such as smoothness and effectiveness, the crucial point in the definition ofa torus manifold is the non-emptiness of the fixed point set. Torus manifolds alsoadmit a combinatorial treatment similar to that of classic toric varieties in terms offans and polytopes. Namely, torus manifolds may be described by multi-fans andmulti-polytopes ; a multi-fan is a collection of cones parametrised by a simplicialcomplex, where some cones may overlap unlike in the usual fan. The cohomologyring of a torus manifold has a more complicated structure than that of a (quasi)toricmanifold; in particular, it is no longer generated by two-dimensional classes. Facerings of simplicial posets (as described in Section 3.5) arise naturally in this context.

Another interesting generalisation of toric manifolds was suggested in thework [170] of Ishida, Fukukawa and Masuda under the name topological toric mani-folds. The idea is to consider not only the actions of a compact torus, but rather two

245

246 7. HALF-DIMENSIONAL TORUS ACTIONS

commuting actions of T n and Rn>, which patch together to a smooth (C×)n-actionon a manifold. This smooth (C×)n-action is what replaces the algebraic (C×)n-action on a toric manifold. The resulting class of manifolds, when properly defined,turns out to be both wide and tractable, and admits a combinatorial description interms of (generalised) fans in a way similar to toric varieties. The topological char-acteristics of topological toric manifolds, including their integral cohomology ringsand characteristic classes, also have much similarity with those of toric manifolds.

In Section 7.1 we discuss the notion of a locally standard torus action and re-lated combinatorics of orbits; it features in many subsequent generalisations of toricmanifolds. Section 7.2 is a brief account of topological properties of the (compact)torus action on toric manifolds, these properties are taken as the base for subse-quent topological generalisations. Sections 7.3, 7.4 and 7.5 describe the classes ofquasitoric manifolds, torus manifolds and topological toric manifolds, respectively.Section 7.4 can be also viewed as an account on topological properties of locally stan-dard T n-manifolds, because the existence of a fixed point (required in the definitionof a torus manifold) often comes as a consequence of other topological restrictionson the locally standard T n-manifold M ; for example, a fixed point automaticallyexists when the odd-degree cohomology of M vanishes. Similarly, a torus manifoldM with Hodd(M) = 0 is necessarily locally standard. In Section 7.6 we describethe relationship between the different classes of half-dimensional torus actions. InSection 7.7 we discuss an important class of examples of projective toric manifoldsobtained as spaces of bounded flags in a complex space. These manifolds illustratenicely many previous constructions with toric and quasitoric manifolds, they willalso feature in the last chapter on toric cobordism. Another class of examples,Bott towers, is the subject of Section 7.8. The study of Bott towers has become animportant part of toric topology, and many interesting open questions arise here.In the last Section 7.9 we explore connections with another active area, the theoryof GKM-manifolds and GKM-graphs, and also study blow-ups of T -manifolds andtheir related combinatorial objects. As usual, more specific introductory remarksare available at the beginning of each section.

7.1. Locally standard actions and manifolds with corners

We have collected here background material on locally standard Tn- and Zn2 -actions and combinatorial structures on their orbit spaces. The latter include thenotions of manifolds with faces and manifolds with corners, which have been stud-ied in differential topology since 1960s in the works of Janich [175], Bredon [41],Davis [88], Davis–Januszkiewicz [90], Izmestiev [174], among others.

As usual, we denote by Z2 (respectively, by S or T) the multiplicative groupof real (respectively, complex) numbers of absolute value one. In this section wedenote by G one of the groups Z2 or S = T, and denote by F its ambient field R orC respectively. We refer to the coordinatewise action of Gn on Fn given by

(g1, . . . , gn) · (z1, . . . , zn) = (g1z1, . . . , gnzn)

as the standard action or standard representation.

Definition 7.1.1. Let M be a manifold with an action of Gn. A standardchart on M is a triple (U, f, ψ), where U ⊂ M is a Gn-invariant open subset, ψ isan automorphism of Gn, and f is a ψ-equivariant homeomorphism f : U →W ontoa Gn-invariant open subset W ⊂ Fn. (Recall from Appendix B.3 that the latter

7.1. LOCALLY STANDARD ACTIONS 247

means that f(t · y) = ψ(t)f(y) for all t ∈ Gn, y ∈ U .) A Gn-action on M is saidto be locally standard if M has a standard atlas, i.e. if any point of M belongs toa standard chart.

The dimension of a manifold with a locally standard Gn-action is n if G = Z2

and 2n if G = T. The orbit space of the standard Gn-action on Fn is the orthant

Rn> =x = (x1, . . . , xn) ∈ Rn : xi > 0 for i = 1, . . . , n

.

Therefore, the orbit space of a locally standard action is locally modelled by Rn>.There are two slightly different ways to formalise this property, depending onwhether we work in the topological or smooth category.

Definition 7.1.2. A manifold with faces (of dimension n) is a topologicalmanifold Q with boundary ∂Q together with a covering of ∂Q by closed connectedsubsets Fii∈S , called facets, satisfying the following properties:

(a) any facet Fi is an (n−1)-dimensional submanifold (with boundary) of ∂Q;(b) for any finite subset I ⊂ S, the intersection

⋂i∈I Fi is either empty or

a disjoint union of submanifolds of codimension |I|; in the latter case werefer to a connected component of the intersection

⋂i∈I Fi as a face of Q;

(c) for any point q ∈ Q, there exist an open neighbourhood U ∋ q and ahomeomorphism ϕ : U →W onto an open subset W ⊂ Rn> such that

ϕ−1(W ∩ xj = 0) = U ∩ Fifor some i ∈ S.

Observe that the set S has to be countable, hence the set of faces in a manifoldwith faces Q is also countable. If Q is compact, then both these sets are finite.

The orthant Rn> itself has the canonical structure of a manifold with faces; each

face has the form RI> for I ⊂ [n], where RI> = x ∈ Rn> : xj = 0 for j /∈ I. The

codimension c(x ) of point x ∈ Rn> is the number of zero coordinates of x . A pointof Q has codimension k if it belongs to a face of codimension k and does not belongto a face of codimension k + 1.

Definition 7.1.3. A manifold with corners (of dimension n) is a topologicalmanifold Q with boundary together with an atlas Ui, ϕi consisting of homeomor-phisms ϕi : Ui →Wi onto open subsets Wi ⊂ Rn> such that ϕiϕ

−1j : ϕj(Ui ∩ Uj)→

ϕi(Ui ∩ Uj) is a diffeomorphism for all i, j. (A homeomorphism between opensubsets in Rn> is called a diffeomorphism if it can be obtained by restriction of a

diffeomorphism of open subsets in Rn.)For any q ∈ Q, its codimension c(q) is well defined. An open face of Q of

codimension k is a connected component of c−1(k). A closed face (or simply face)of Q is the closure of an open face. A facet is a face of codimension 1.

A manifold with corners Q is said to be nice if the covering of Q by its facetssatisfies condition (c) of Definition 7.1.2 (conditions (a) and (b) are satisfied auto-matically). Equivalently, a manifold with corners is nice if and only if each of itsfaces of codimension 2 is contained in exactly 2 facets (an exercise).

Remark. Our definitions of faces differ from those of [88] and [174]; the reasonis that we want our faces to be connected.


Example 7.1.4.1. The 2-disc with a single ‘corner point’ on its boundary is a manifold with

corners which is not nice. All other examples in this list will be nice.2. A smooth manifold Q with boundary is a manifold with corners, whose

facets are connected components of ∂Q, and there are no other faces.3. A direct product of manifolds with corners is a manifold with corners. In

particular, a product of smooth manifolds with boundary is a manifold with corners.4. Let P be a simple polytope. For each vertex v ∈ P we denote by Uv the

open subset in P obtained by removing all faces of P that do not contain v. Thesubset Uv is affinely isomorphic to a neighbourhood of zero in Rn>. Therefore P is

a compact manifold with corners, with atlas Uv.The orbit space Q = M/Gn of a locally standard action is a manifold with

faces. As we shall see in Proposition 7.4.13, if the action is smooth, then Q is anice manifold with corners.

Exercises.

7.1.5. A manifold with corners Q is nice if an only if any face of codimensiontwo is contained in exactly two facets.

7.1.6. Two simple polytopes are diffeomorphic as manifolds with corners if andonly if they are combinatorially equivalent (see [89] for a more general statement).

7.2. Toric manifolds and their quotients

By way of motivation, here we take a closer look at the action of the (compact)torus TN ∼= T n on a toric manifold VΣ. The topological properties of the quotientprojection π : VΣ → VΣ/TN will be taken as the starting point for subsequenttopological generalisations of toric manifolds.

We first recall from Sections 5.5 and 6.3 that a projective (or Hamiltonian) toricmanifold VP can be identified with the symplectic quotient of Cm by an action ofK ∼= Tm−n, i.e. with the quotient manifold ZP /K where ZP is the moment-anglemanifold corresponding to P . Therefore the quotient of VP by the action of then-torus TN = Tm/K coincides with the quotient of ZP by Tm. Both quotients areidentified with the Delzant polytope P ; in fact, the moment map µV : VP → P is thequotient projection (see Proposition 5.5.5). In the non-projective smooth case, thereis no moment map, and there is no canonical way to identify the quotient VΣ/TNwith a convex polytope. However, there is a face decomposition (stratification) ofVΣ/TN according to orbit types, and this face structure is very similar to that of asimple polytope.

Following Davis–Januszkiewicz [90], we can describe a projective toric manifoldVP as an identification space similar to (6.7).

As usual, for each I ⊂ [m] we denote by TI =∏i∈I T the corresponding

coordinate subgroup in Tm. Given x ∈ P , set Ix = i ∈ [m] : x ∈ Fi (the set offacets containing x). We recall the map A : Rm → NR, e i 7→ a i, and its exponentialexpA : Tm → TN . For each x ∈ P define the subtorus Tx = (expA)(TIx ) ⊂ TN . Ifx is a vertex then Tx = TN , and if x is an interior point of P then Tx = 1.

Proposition 7.2.1. A projective toric manifold VP is TN -equivariantly home-omorphic to the quotient

(7.1) P × TN/≈ where (x, t1) ≈ (x, t2) if t−11 t2 ∈ Tx.

7.3. QUASITORIC MANIFOLDS 249

Proof. Using Proposition 6.2.2, we obtain

VP = ZP /K =(P × (Tm/K)

)/∼ =

(P × (expA)(Tm)

)/∼ = P × TN/≈.

The projection π : VP = P ×TN/≈ → P is the quotient map for the TN -action,and its fibre π−1(x ) = Tx is the stabiliser of the TN -orbit corresponding to x .The TN -action on VP is therefore free over the interior of the polytope, verticesof the polytope correspond to fixed points, and points in the relative interior of acodimension-k face correspond to orbits with the same k-dimensional stabiliser.

Construction 7.2.2. Let Σ be a simplicial fan and VΣ the corresponding toricvariety. Consider the affine cover Vσ : σ ∈ Σ (see Construction 5.1.3). The quo-tient (Vσ)> = Vσ/TN can be identified with the set of semigroup homomorphismsfrom Sσ to the semigroup R> of nonnegative real numbers:

Vσ/TN = Homsg(Sσ,R>), σ ∈ Σ

(the details of this construction can be found in [122, §4.1]).If the fan Σ is regular, then Vσ ∼= Ck × (C×)n−k where k = dimσ, see Ex-

ample 5.1.5. It follows easily that the TN -action on a nonsingular toric varietyVΣ is locally standard, and the cover Vσ : σ ∈ Σ provides an atlas of standardcharts (see Section 7.1). Furthermore, (Vσ)> ∼= Rk> × Rn−k and the orbit space

Q = VΣ/TN is a manifold with corners, with atlas (Vσ)> : σ ∈ Σ.In the singular case the varieties Vσ may be not isomorphic to Ck × (C×)n−k

and the TN -action on VΣ may fail to be locally standard. Nevertheless, the cover(Vσ)> : σ ∈ Σ defines a structure of a manifold with faces on Q = VΣ/TN .

If Σ is a complete simplicial fan, then the face decomposition of the manifoldwith corners Q = VΣ/TN is Poincare dual to the sphere triangulation defined bythe simplicial complex KΣ. There is also a projection Q × TN → VΣ defining ahomeomorphism VΣ ∼= Q × TN/≈, by analogy with (7.1).

We finish by summarising the observations of this section as follows:

Proposition 7.2.3.

(а) The action of the torus TN on a nonsingular toric variety VΣ is locallystandard.

(b) If VP is a projective toric manifold then the quotient VP /TN is diffeomor-phic to the simple polytope P as a manifold with corners.

If a toric manifold VΣ is not projective, then the manifold with corners Q =VΣ/TN may be not homeomorphic to a simple polytope (see Section 7.6).

7.3. Quasitoric manifolds

Quasitoric manifolds were introduced by Davis and Januszkiewicz as a topo-logical alternative to (nonsingular projective) toric varieties. Originally, the term‘toric manifolds’ was used in [90] to describe this class of manifolds, but later is wasreplaced by ‘quasitoric’, as ‘toric manifold’ is often used by algebraic and symplecticgeometers as a synonym for ‘non-singular complete toric variety’.

Any quasitoric manifold over P can be obtained as a quotient of the moment-angle manifold ZP by a freely acting subtorus. This can be viewed as a topologicalversion of the symplectic quotient construction of projective toric manifolds (seeSection 5.5). As a result we obtain a canonical smooth structure on a quasitoricmanifold, in which the torus action is smooth.


Unlike toric manifolds, quasitoric manifolds are not complex varieties in general,and they may even not admit an almost complex structure. However, an equivariantstably complex structure always exists on a quasitoric manifold M and is definedcanonically by the underlying combinatorial data. These structures will be used inChapter 9 to define quasitoric representatives in complex cobordism classes.

Definition and basic constructions.

Definition 7.3.1. Let P be a combinatorial simple polytope of dimension n. Aquasitoric manifold over P is a smooth 2n-dimensional manifold M with a smoothaction of the torus T n satisfying the two conditions:

(a) the action is locally standard (see Definition 7.1.1);(b) there is a continuous projection π : M → P whose fibres are T n-orbits.

Property (a) implies that the quotientM/P is a manifold with corners, and property(b) implies that the quotient is homeomorphic, as a manifold with corners, to thesimple polytope P .

It follows from the definition that the projection π : M → P maps a k-dimensional orbit of the T n-action to a point in the relative interior of a k-dimensional face of P . In particular, the action is free over the interior of thepolytope, while vertices of P correspond to fixed points of the torus action on M .

Proposition 7.3.2. A nonsingular projective toric variety VP is a quasitoricmanifold over P .

Proof. This follows from Proposition 7.2.3.

Example 7.3.3. The complex projective space CPn with the action of T n ⊂(C×)n described in Example 5.1.2 is a quasitoric manifold over the simplex ∆n.The projection π : CPn → ∆n is given by

(z0 : z1 : · · · : zn) 7→1∑n

i=0 |zi|2(|z1|2, . . . , |zn|2).

Remark. For any 2- or 3-dimensional polytope P there exists a quasitoricmanifold over P . For n > 4, there exist n-dimensional polytopes which do not ariseas quotients of quasitoric manifolds. (See Exercises 7.3.29 and 7.3.30.)

Let F = F1, . . . , Fm be the set of facets of P . Consider the preimages

Mj = π−1(Fj), 1 6 j 6 m.

Points in the relative interior of a facet Fj correspond to orbits with the same one-dimensional stabiliser subgroup, which we denote by TFj

. It follows that Mj is aconnected component of the fixed point set of the circle subgroup TFj

⊂ T n. Thisimplies that Mj is a T n-invariant submanifold of codimension 2 in M , and Mj is aquasitoric manifold over Fj with the action of the quotient torus T n/TFj

∼= T n−1.Following [90], we refer to Mj as the characteristic submanifold corresponding tothe jth face Fj ⊂ P . The mapping

(7.2) λ : Fj 7→ TFj, 1 6 j 6 m,

is called the characteristic function of the quasitoric manifold M .Now let G be a codimension-k face of P . We can write it as an intersection of

k facets: G = Fj1 ∩ · · · ∩Fjk . Then MG = π−1(G) is a T n-invariant submanifold ofcodimension 2k inM , andMG is fixed under each circle subgroup T (Fjp), 1 6 p 6 k.


By considering any vertex v ∈ G and using the local standardness of the T n-actiononM near v, we observe that the characteristic submanifoldsMj1 , . . . ,Mjk intersecttransversely at the submanifold MG, and the map

TFj1× · · · × TFjk

→ T n

is a monomorphism onto the k-dimensional stabiliser of MG. The mapping

G 7→ the stabiliser of MG

extends the characteristic function (7.2) to a map from the face poset of P to theposet of torus subgroups in T n.

Definition 7.3.4. Let P be a combinatorial n-dimensional simple polytopeand let λ be a map from the set of facets of P to the set of circle subgroups ofthe torus T n. We refer to (P, λ) as a characteristic pair if the map λ(Fj1 )× · · · ×λ(Fjk )→ T n is a monomorphism whenever Fj1 ∩ · · · ∩ Fjk 6= ∅.

If (P, λ) is a characteristic pair, then the map λ extends to the face poset of P ,and we have a torus subgroup TG = λ(G) ⊂ T n for each face G ⊂ P .

As we shall see below, a quasitoric manifold can be reconstructed from itscharacteristic pair (P, λ) up to a weakly T -equivariant homeomorphism.

Construction 7.3.5 (canonical model M(P, λ)). Assume given a characteris-tic pair (P, λ). For any point x ∈ P , we denote by G(x) the smallest face contain-ing x. By analogy with (7.1), we define the identification space

M(P, λ) = P × T n/∼ where (x, t1) ∼ (x, t2) if t−11 t2 ∈ λ(G(x)).

The free action of T n on P ×T n descends to an action on P ×T n/∼. This action isfree over the interior of the polytope (since no identifications are made over intP ),and the fixed points correspond to the vertices. The space P × T n/∼ is covered bythe open subsets Uv × T n/∼, indexed by the vertices v ∈ P (see Example 7.1.4.4),and each Uv × T n/∼ is equivariantly homeomorphic to Cn = Rn> × T n/∼. This

implies that the canonical model M(P,Λ) is a (topological) manifold with a locallystandard T n-action and quotient P . As we shall see from Definition 7.3.13, M(P,Λ)has a canonical smooth structure, so it is a quasitoric manifold over P .

Proposition 7.3.6 ([90, Lemma 1.4]). There exists a weakly T n-equivarianthomeomorphism

M(P, λ) = P × T n/∼ →M

covering the identity map on P .

Proof. One first constructs a weakly T n-equivariant map f : P × T n → Msuch that f maps x× T n onto π−1(x) for any point x ∈ P . Such a map f induces

a weakly T n-equivariant map f : M(P, λ) = P × T n/∼ →M covering the identity

on P . Furthermore, the map f is one-to-one on T n-orbits, so it is a homeomorphism.It remains to construct a map f : P × T n → M . The argument of Davis–

Januszkiewicz which we present here actually works for a more general class of

locally standard T n-manifoldsM . There is the manifold with boundary M obtainedby consecutive blowing up the singular strata of M consisting of non-principal T n-

orbits. The T n-action on M is free and M is equivariantly diffeomorphic to thecomplement in M of the union of tubular neighbourhoods of the singular strata(the latter are characteristic submanifolds Mj in our case). There is the following


canonical inductive procedure for constructing M from M . One begins by removinga minimal stratum (a fixed point in our case) from M and replacing it by the spherebundle of its normal bundle. One continues in this fashion, blowing up minimalstrata, until only top stratum is left. There is the canonical projection from theunion of sphere bundles to the union of their base spaces (i.e. to the union ofsingular strata of M). Using the construction of [87, p. 344] one extends this

projection to a map M → M which is the identity over top stratum. Now if M is

locally standard, then the quotient M/T n is canonically identified with M/T n. Inour particular case the latter is a simple polytope P . Since P is acyclic together

with all its faces, the resulting principal T n-bundle π : M → P is trivial. Therefore,

there is an equivariant diffeomorphism f : P ×T n → M inducing the identity on P .

Composing f with the collapse map M →M , we obtain the map f .

Remark. Note that existence a map f : P × T n → M in the proof above isequivalent to existence of a section s : P →M of the quotient projection π : M → P .Indeed, given a section s, one defines f(x, t) = t · s(x) for x ∈ P and t ∈ T n.Conversely, given a map f , one defines a section by s(x) = f(x, 1).

It may seem that constructing a section s : P → M is an easier task thanconstructing a map f , taking into account that P is contractible. However, thiscomes out to be subtle; it would be interesting to find a more explicit way toconstruct a section s.

Definition 7.3.7 (equivalences). Quasitoric manifolds M1 and M2 over thesame polytope P are said to be equivalent over P if there exists a weak T n-equivariant homeomorphism f : M1 →M2 covering the identity map on P .

Two characteristic pairs (P, λ1) and (P, λ2) are said to be equivalent if thereexists an automorphism ψ : T n → T n such that λ2 = ψ · λ1.

Proposition 7.3.8 ([90, Proposition 1.8]). There is a one-two-one correspon-dence between equivalence classes of quasitoric manifolds and characteristic pairs.In particular, for any quasitoric manifold M over P with characteristic function λ,there is a homeomorphism M ∼=M(P, λ).

Proof. Obviously, if two quasitoric manifolds are equivalent over P , thentheir characteristic pairs are also equivalent. To establish the other implication,it is enough to show that a quasitoric manifold M is equivalent to the canonicalmodel M(P, λ). This follows from Proposition 7.3.6.

Omniorientations and combinatorial quasitoric data. Here we elaborateon the combinatorial description of quasitoric manifolds M . Characteristic pairs(P, λ) are replaced by more naturally defined combinatorial quasitoric pairs (P,Λ)consisting of an oriented simple polytope and an integer matrix of special type.Compared with the characteristic pair, the pair (P,Λ) carries some additional in-formation, which is equivalent to a choice of orientation for the manifold M andits characteristic submanifolds. The terminology and constructions described herewere introduced in [62] and [58].

Definition 7.3.9. An omniorientation of a quasitoric manifold M consists ofa choice of orientation for M and each characteristic submanifold Mj, 1 6 j 6 m.

In general, an omniorientation on M cannot be chosen canonically. However,if M admits a T n-invariant almost complex structure (see Definition D.6.1), then


a choice of such structure provides canonical orientations for M and the invariantsubmanifolds Mj . We therefore obtain an omniorientation associated with theinvariant almost complex structure. In the case when M has an invariant almostcomplex structure (for example, when M is a toric manifold), we always choose theassociated omniorientation. Otherwise we choose an omniorientation arbitrarily.

The stabiliser TFjof a characteristic submanifold Mj ⊂M can be written as

(7.3) TFj=(e2πiλ1jϕ, . . . , e2πiλnjϕ

)∈ Tn

,

where ϕ ∈ R и λj = (λ1j , . . . , λnj)t ∈ Zn is a primitive vector. (In the coordinate-

free notation used in Chapter 5, the vector λj belongs to the lattice N of one-parameters subgroups of the torus.) This vector is determined by the subgroupTFj

up to sign. A choice of this sign (and therefore an unambiguous choice of thevector) defines a parametrisation of the circle subgroup TFj

.An omniorientation of M provides a canonical way to choose the vectors λj .

Indeed, the action of a parametrised circle TMj⊂ Tn defines an orientation in the

normal bundle νj of the embedding Mj ⊂ M . An omniorientation also defines anorientation on νj by means of the following decomposition of the tangent bundle:

TM |Mj= TMj ⊕ νj .

Now we choose the direction of the primitive vector λj so that these two orientationscoincide.

Having fixed an omniorientation, we can extend correspondence (7.2) to a mapof lattices

Λ : Zm → Zn, ej 7→ λj ,

which we refer to as a directed characteristic function. A characteristic function isassumed to be directed whenever an omniorientation is chosen.

We can think of a directed characteristic function as an integer n×m-matrix Λwith the following property: if the intersection of facets Fj1 , . . . , Fjk is nonempty,then the vectors λj1 , . . . , λjk form a part of basis of the lattice Zn. We refer tosuch matrices as characteristic. In particular, we can write any vertex v ∈ P asan intersection of n facets: v = Fj1 ∩ · · · ∩ Fjn , and consider the maximal minorΛv = Λj1,...,jn formed by the columns j1, . . . , jn of matrix Λ. Then

(7.4) detΛv = ±1.Since M ∼=M(P, λ) = P×T n/∼, and no identifications are made over the inte-

rior of P , a choice of an orientation for M is equivalent to a choice of an orientationfor the polytope P (once we assume that the torus T n is oriented canonically).

Definition 7.3.10. Let P be an oriented combinatorial simple n-polytope withm facets, and let Λ be an integer n ×m-matrix satisfying condition (7.4) for anyvertex v ∈ P . Then (P,Λ) is called a combinatorial quasitoric pair.

An equivalence of omnioriented quasitoric manifolds is assumed to preserve theomniorientation; in this case the automorphism ψ : T n → T n in Definition 7.3.7is orientation-preserving. Similarly, two combinatorial quasitoric pairs (P,Λ1) and(P,Λ2) are said to be equivalent if the orientation of P coincide and there exists asquare integer matrix Ψ with determinant 1 such that Λ2 = Ψ · Λ1.

We can summarise the observations above in the following refined version ofProposition 7.3.8:


Proposition 7.3.11. There is a one-to-one correspondence between equivalenceclasses of omnioriented quasitoric manifolds and combinatorial quasitoric pairs.

We shall denote the omnioriented quasitoric manifold corresponding to a com-binatorial quasitoric pair (P,Λ) by M(P,Λ).

Let chf(P ) denote the set of directed characteristic functions λ for P . Thegroup GL(n,Z) of automorphisms of the torus T n acts on the set chf(P ) from theleft. Proposition 7.3.11 establishes a one-two-one correspondence

(7.5) GL(n,Z)\ chf(P )←→ equivalence classes of omnioriented M over P.If the facets of P are ordered in such a way that the first n of them meet at a

vertex v, i.e. F1 ∩ · · · ∩ Fn = v, then each coset from GL(n,Z)\ chf(P ) contains aunique directed characteristic function given by a matrix of the form

(7.6) Λ = (I | Λ⋆) =

1 0 . . . 0 λ1,n+1 . . . λ1,m0 1 . . . 0 λ2,n+1 . . . λ2,m...

.... . .

......

. . ....

0 0 . . . 1 λn,n+1 . . . λn,m

where where I is the unit matrix and Λ⋆ is an n × (m − n)-matrix. We referto (7.6) as a refined characteristic matrix, and to Λ∗ as its refined submatrix. If acharacteristic matrix is given in a non-refined form Λ = (A | B), where A has sizen× n, then its refined representative is given by (I | A−1B).

Smooth and stably complex structures. Let (P,Λ) be a combinatorialquasitoric pair. The matrix Λ defines an epimorphism of tori expΛ : Tm → Tn,whose kernel we denote by K = K(Λ). Condition (7.4) implies that K ∼= Tm−n.We therefore have an exact sequence of tori similar to (5.9). Also, there is themoment-angle manifold ZP corresponding to the polytope P (see Section 6.2).

Proposition 7.3.12. The groupK(Λ) ∼= Tm−n acts freely and smoothly on ZP .There is a T n-equivariant homeomorphism

ZP /K(Λ)∼=−→M(P,Λ)

between the quotient ZP /K(Λ) and the canonical model M(P,Λ).

Proof. The fact that K acts freely on ZP is proved in the same way as Propo-sition 5.4.6 (a) and Theorem 6.5.2 (a). The stabiliser of a point z ∈ ZP with respectto the Tm-action is a coordinate subtorus TI for some I ∈ KP . Namely, if z ∈ ZPprojects to x ∈ P , then I = Ix = i ∈ [m] : x ∈ Fi. Condition (7.4) implies thatthe restriction of the homomorphism expΛ : Tm → Tn to any such subtorus TI isinjective, and therefore the kernel K intersects each TI , I ∈ KP , trivially. HenceK acts freely on ZP , and the quotient ZP /K is a 2n-dimensional manifold with anaction of the torus Tm/K ∼= Tn.

To prove the second statement, we identify ZP with the quotient P × Tm/∼,see Section 6.2. Then the projection ZP → ZP /K is identified with the projection

ZP = P × Tm/∼ → P × Tn/∼,induced by the homomorphism expΛ : Tm → Tn. By Construction 7.3.5, the quo-tient P × Tn/∼ is the canonical quasitoric manifold M(P,Λ).


Definition 7.3.13. Using Proposition 7.3.12 we obtain a smooth structure onthe canonical model M(P,Λ) as the quotient of the smooth manifold ZP by thesmooth action of K(Λ), with the induced smooth action of the n-torus Tm/K(Λ).We refer to this smooth structure on M(P,Λ) and on any quasitoric manifoldequivalent to it as canonical.

Now let p : ZP → P be the projection, and consider the submanifolds p−1(Fi) ⊂ZP corresponding to facets Fi of P (see Exercise 6.2.13). The submanifold p−1(Fi)is fixed by the ith coordinate subcircle in Tm. Denote by Ci the space of the1-dimensional complex representation of the torus Tm induced from the standardrepresentation in Cm by the projection Cm → Ci onto the ith coordinate. LetZP ×Ci → ZP be the trivial complex line bundle; we view it as an equivariant Tm-bundle with diagonal action of Tm. Then the restriction of the bundle ZP × Ci →ZP to the invariant submanifold p−1(Fi) is Tm-isomorphic to the normal bundleof the embedding p−1(Fi) ⊂ ZP . By taking quotient with respect to the diagonalaction of K = K(Λ) we obtain a T n-equivariant complex line bundle

(7.7) ρi : ZP ×K Ci → ZP /K =M(P,Λ)

over the quasitoric manifold M =M(P,Λ). The restriction of the bundle ρi to thecharacteristic submanifold p−1(Fi)/K =Mi is isomorphic to the normal bundle ofMi ⊂ M (an exercise). The resulting complex structure on this normal bundle isthe one defined by the omniorientation of M(P,Λ).

Theorem 7.3.14 ([90, Theorem 6.6]). There is the following isomorphism ofreal T n-bundles over M =M(P,Λ):

(7.8) TM ⊕ R2(m−n) ∼= ρ1 ⊕ · · · ⊕ ρm;

here R2(m−n) denotes the trivial real 2(m− n)-dimensional T n-bundle over M .

Proof. The proof given here differs from that of [90]: we use the equivariantframing of ZP coming from its realisation by an intersection of quadrics, see Sec-tion 6.2. Let iZ : ZP → Cm be the Tm-equivariant embedding, see (6.2). We havea Tm-equivariant decomposition

(7.9) T ZP ⊕ ν(iZ) = ZP × Cm

obtained by restricting the tangent bundle T Cm to ZP . The normal bundle ν(iZ)is Tm-equivariantly trivial by Theorem 6.1.3, and ZP × Cm is isomorphic, as aTm-bundle, to the sum of line bundles ZP × Ci, 1 6 i 6 m. Further, we have

(7.10) T ZP = q∗(TM)⊕ ξ,where ξ is the tangent bundle along the fibres of the principal K-bundle q : ZP →ZP /K = M , see [300, Corollary 6.2]. The bundle ξ is induced by the projectionq from the vector bundle over M associated with the principal bundle ZP → Mthrough the adjoint representation of the Lie group K; since K is abelian, this bun-dle is trivial. Taking quotient of identity (7.9) by the action of K and using (7.10),we obtain a decomposition

(7.11) TM ⊕ (ξ/K)⊕ (ν(iZ)/K) ∼= ZP ×K Cm.

As we have seen above, both ξ and ν(iZ) are trivial Tm-bundles, so that (ξ/K)⊕(ν(iZ)/K) ∼= R2(m−n). Also, ZP ×K Cm ∼= ρ1 ⊕ · · · ⊕ ρm as T n-bundles.


The isomorphism of Theorem 7.3.14 gives us an isomorphism of the stabletangent bundle of M with a complex vector bundle: this is the setup for a stablycomplex structure (see Definition D.6.1).

Corollary 7.3.15. An omnioriented quasitoric manifold M has a canonicalT n-invariant stably complex structure cT defined by the isomorphism of (7.8).

The corresponding bordism classes [M ] ∈ ΩU2n will be studied in Section 9.5.

Example 7.3.16. Let us see which stably complex structures we can obtainfrom different omniorientations on the simplest quasitoric manifold S2 over thesegment I1 using Theorem 7.3.14. The standard complex structure of CP 1 hasthe characteristic matrix (1 −1). The group K is the diagonal circle (t, t) ⊂ T2,and (7.11) becomes the standard decomposition

T CP 1 ⊕ C ∼= S3 ×K C2 = η ⊕ η,where η is the tautological and η is the canonical line bundle, see Exercise B.3.6.

On the other hand, there is an omniorientation of S2 corresponding to thecharacteristic matrix (1 1). Then K = (t−1, t) ⊂ T2, and (7.11) becomes

T S2 ⊕ R2 ∼= S3 ×K C2 = η ⊕ η,which is the trivial stably complex structure on S2 (see Example D.3.1).

Weights and signs at fixed points. Any fixed point v of the T n-action on aquasitoric manifold M is isolated. It can be obtained as an intersection Mj1 ∩ · · · ∩Mjn of n characteristic submanifolds and corresponds to a vertex Fj1 ∩ · · · ∩Fjn ofthe polytope P . Therefore, the tangent space to M at v decomposes into the sumof normal spaces to Mjk for 1 6 k 6 n:

(7.12) TvM = (ρj1 ⊕ · · · ⊕ ρjn)|v.We use this decomposition to identify TvM with Cn; then the tangent space to Mjk

is given in the corresponding coordinates (z1, . . . , zn) by the equation zk = 0. Therepresentation of the torus T n in the tangent space TvM ∼= Cn is determined by itsset of weights wk(v) ∈ Zn, 1 6 k 6 n. Namely, for t = (e2πiϕ1 , . . . , e2πiϕn) ∈ T nand z = (z1, . . . , zn) ∈ TvM , we have

t ·z = (e2πi〈w1(v), ϕ〉z1, . . . , e2πi〈wn(v), ϕ〉zn),

where ϕ = (ϕ1, . . . , ϕn) ∈ Rn. The weights can be found from the combinatorialquasitoric pair (P,Λ) using the following statement:

Proposition 7.3.17. Let M =M(P,Λ) be a quasitoric manifold. The weightsw1(v), . . . ,wn(v) of the tangent representation of T n at a fixed point v = Mj1 ∩· · · ∩Mjn are given by the columns of the square matrix Wv satisfying the identity

W tv Λv = I.

In other words, w1(v), . . . ,wn(v) is the lattice basis conjugate to λj1 , . . . , λjn.Proof. First, note that the local standardness of the action implies that

w1(v), . . . ,wn(v) is a lattice basis. (The fact that λj1 , . . . , λjn is a latticebasis is expressed by identity (7.4).)

Since the one-parameter subgroup TFjk⊂ T n (see (7.3)) fixes the hyperplane

zk = 0 tangent to Mjk , we obtain that 〈w i(v), λjk 〉 = 0 for i 6= k. Therefore, W tvΛv


is a diagonal matrix. Now, the columns of both Wv and Λv are lattice bases, whichimplies 〈wk(v), λjk 〉 = ±1 for 1 6 k 6 n.

On the other hand, the complex structure on the line bundle ρjk comes fromthe orientation induced by the action of the one-parameter subgroup of T n corre-sponding to the vector λjk . Hence 〈wk(v), λjk 〉 > 0 for 1 6 k 6 n.

The signs of the fixed points defined by the T n-invariant stably complex struc-ture on M (see Definition D.6.2) can be calculated in terms of the combinatorialquasitoric pair (P,Λ) as follows:

Lemma 7.3.18. Let v =Mj1 ∩ · · · ∩Mjn be a fixed point.

(a) In terms of decomposition (7.12), we have σ(v) = 1 if the orientationof the space TvM determined by the orientation of M coincides with theorientation of the space (ρj1 ⊕ · · · ⊕ ρjn)|v determined by the orientationof the line bundles ρjk , 1 6 k 6 n. Otherwise, σ(v) = −1.

(b) In terms of the combinatorial quasitoric pair (P,Λ), we have

σ(v) = sign(det(λj1 , . . . , λjn) det(aj1 , . . . ,ajn)

),

where aj1 , . . . ,ajn are inward-pointing normals to the facets Fj1 , . . . , Fjn .

Proof. To prove (a), we note that the complex line bundle ρi is trivial over thecomplement to Mi in M . Therefore, the nontrivial part of the T n-representation(ρ1⊕ · · ·⊕ ρm)|v is exactly (ρj1 ⊕ · · ·⊕ ρjn)|v. This implies that the composite mapin Definition D.6.2 is given by

(7.13) TvM → (ρj1 ⊕ · · · ⊕ ρjn)|v.To prove (b), we write map (7.13) in coordinates. To do this, we identify Cm

with R2m by mapping a point (z1, . . . , zm) ∈ Cm to (x1, . . . , xm, y1, . . . , ym) ∈ R2m,where zk = xk + iyk. Using decomposition (7.11), we obtain that the map

TvM → TvM ⊕ R2(m−n) cT−→ (ρ1 ⊕ · · · ⊕ ρm)|v ∼= R2m

from Definition D.6.2 is given by the 2m× 2n-matrix(At 0

0 Λt

)

where A denotes the n ×m-matrix with columns vectors a i defined by presenta-tion (1.1) of the polytope P . Map (7.13) is obtained by restricting to the submatri-ces of At and Λt formed by rows with numbers j1, . . . jn, which implies the requiredformula for the sign.

Example 7.3.19. Let VP be the projective toric manifold corresponding to asimple lattice polytope P given by (1.1). Then λi = ai for 1 6 i 6 m. Proposi-tion 7.3.17 implies that the weights w1(v), . . . ,wn(v) of the tangent representationof the torus at a fixed point v ∈ VP are the primitive vectors along the edges of Ppointing out of v. Furthermore, Lemma 7.3.18 implies that σ(v) = 1 for all v.

For a general quasitoric manifold, the weights w1(v), . . . ,wn(v) are not vec-tors along edges. However, by Proposition 7.3.17, there is a natural one-to-onecorrespondence

(7.14) oriented edges of P↔ weights of the tangential T n-representations at fixed points.


Under this correspondence, an edge e coming out of a vertex v = Fj1 ∩· · ·∩Fjn ∈ Pmaps to wk(v), where Fjk is the unique facet containing v and not containing eand wk(v) is the kth vector of the conjugate basis of λj1 , . . . , λjn .

Lemma 7.3.20. Let e1, . . . , en be vectors along the edges coming out of a ver-tex v ∈ P , and let w1(v), . . . ,wn(v) be the corresponding weights. Then there is thefollowing formula for the sign of the vertex:

σ(v) = sign(det(w1(v), . . . ,wn(v)) det(e1, . . . , en)

).

Proof. This follows from Lemma 7.3.18 and the fact that w1(v), . . . ,wn(v)is the conjugate basis of λj1 , . . . , λjn, while the vectors e1, . . . , en can be scaledso that they form the conjugate basis of aj1 , . . . ,ajn.

Remark. Formulae of Lemmata 7.3.18 and 7.3.20 can be rephrased as thefollowing practical rule for calculation of signs, which will be used below. Writethe vectors λj1 , . . . , λjn (respectively, w1(v), . . . ,wn(v)) into a square matrix inthe order given by the orientation of P , that is, in such a way that inward-pointingnormals of the corresponding facets (respectively, vectors along the correspondingedges) form a positive basis of Rn. Then the determinant of this matrix is σ(v).In other words, assuming that the weights w1(v), . . . ,wn(v) are ordered so thatvectors along their corresponding edges form a positive basis, we can rewrite theformula of Lemma 7.3.20 as

(7.15) σ(v) = det(w1(v), . . . ,wn(v)).

Example 7.3.21. The complex projective plane CP 2 has the standard stablycomplex complex structure coming from the bundle isomorphism

T (CP 2)⊕ C ∼= η ⊕ η ⊕ η,where η is the tautological line bundle. The orientation is determined by the com-plex structure. The toric manifold CP 2 corresponds to the lattice 2-simplex ∆2

with vertices (0, 0), (1, 0) and (0, 1). The column vectors λ1, λ2, λ3 of the matrixΛ are the primitive inward-pointing normals to the facets (i.e. they coincide withthe vectors a1,a2,a3 for the standard presentation of ∆2). The weights of thetangential T 2-representation at a fixed point are the primitive vectors along theedges coming out of the corresponding vertex. This is shown in Fig. 7.1. We haveσ(v1) = σ(v2) = σ(v3) = 1.

Example 7.3.22. Now consider CP 2 with the omniorientation defined by thethree vectors λ1, λ2, λ3 in Fig. 7.2. This omniorientation differs from the previousone by the sign of λ3. The stably complex structure is defined by the isomorphism

T (CP 2)⊕ R2 ∼= η ⊕ η ⊕ η.Using formula (7.15), we calculate

σ(v1) =

∣∣∣∣1 00 1

∣∣∣∣ = 1, σ(v2) =

∣∣∣∣−1 11 0

∣∣∣∣ = −1, σ(v3) =

∣∣∣∣0 11 −1

∣∣∣∣ = −1.

If the omniorientation of M comes from a T n-invariant almost complex struc-ture, then all signs of fixed points are positive because the two orientation in (7.12)coincide. As we shall see below in this section, the top Chern number of the stablycomplex structure of an omnioriented quasitoric manifold M is equal to the sum


v1 (1, 0) (−1, 0) v2

λ2 = (0, 1)

(0, 1)

λ1 = (1, 0)

(0,−1)

v3

(1,−1)

(−1, 1)

λ3 = (−1,−1)

Figure 7.1. T (CP 2)⊕ C ∼= η ⊕ η ⊕ η

v1 (1, 0) (1, 0) v2

λ2 = (0, 1)

(0, 1)

λ1 = (1, 0)

(0, 1)

v3

(1,−1)

(−1, 1)

λ3 = (1, 1)

Figure 7.2. T (CP 2)⊕ C ∼= η ⊕ η ⊕ η

of signs of fixed points: cn[M ] =∑

v∈M σ(v). On the other hand, the Euler char-acteristic χ(M) is equal to the number of fixed points. Therefore, the positivity ofall signs is equivalent to the condition cn[M ] = χ(M). According to the classicalresult of Thomas [306], this condition is sufficient for a stably complex manifoldM to be almost complex. The following result of Kustarev shows that this almostcomplex structure can be made T n-invariant:

Theorem 7.3.23 ([190]). A quasitoric manifold M admits a T n-invariant al-most complex structure if and only if it admits an omniorientation in which allsigns of fixed points are positive.

Theorem 7.3.23 allows one to construct an invariantly almost complex qua-sitoric manifold which is not toric (see Exercise 7.3.34 below). Such an almost


complex structure cannot be integrable. Indeed, according to the result of Ishida–Karshon [171] (Theorem 6.6.8, see also [172]), a quasitoric manifold with an in-variant complex structure is biholomorphic to a compact toric variety.

Cohomology ring and characteristic classes. The cohomology ringH∗(M) of a quasitoric manifold M has the same structure as the cohomologyring of a nonsingular compact toric variety (see Theorem 5.3.1). In particular, thering H∗(M) is generated by two-dimensional classes vi dual to the characteristicsubmanifolds (or equivalently, by the first Chern classes of line bundles ρi fromTheorem 7.3.14). The elements vi satisfy two types of relations: monomial rela-tions coming from the face ring of the polytope P and linear relations coming fromthe characteristic matrix Λ.

Let M =M(P,Λ) be an omnioriented quasitoric manifold, and let π : M → Pbe the projection onto the orbit space. We start by describing a canonical celldecomposition of M with only even-dimensional cells. It was first constructed byKhovanskii [183] for toric manifolds.

Construction 7.3.24. Recall the ‘Morse-theoretic’ arguments used in theproof of the Dehn–Sommerville relations (Theorem 1.3.4). There we turned the1-skeleton of P into an oriented graph and defined the index ind(v) of a vertexv ∈ P as the number of incoming edges. The incoming edges of v span a face Gv of

dimension ind(v). Denote by Gv the subset obtained by removing from Gv all faces

not containing v. Then Gv is homeomorphic to Rind(v)> and is contained in the open

set Uv ⊂ P from Example 7.1.4.4. The preimage ev = π−1Gv is homeomorphic toCind(v) and the union of subsets ev ⊂ M over all vertices v ∈ P defines a cell de-composition of M . Observe that all cells have even dimension, and the closure ofthe cell ev is the quasitoric submanifold π−1(Gv) ⊂M .

Proposition 7.3.25. The homology groups of a quasitoric manifold M =M(P,Λ) vanish in odd dimensions, and therefore are free abelian in even dimen-sions. Their ranks (Betti numbers) are given by

b2i(M) = hi(P ),

where hi(P ), i = 0, 1, . . . , n, are the components of the h-vector of P .

Proof. The rank of H2i(M ;Z) is equal to the number of 2i-dimensional cellsin the above cell decomposition. This number is equal to the number of vertices ofindex i, which is hi(P ) by the argument from the proof of Theorem 1.3.4.

Now consider the face ring Z[P ] (see Section 3.1) and define its elements

(7.16) ti = λi1v1 + · · ·+ λimvm ∈ Z[P ], 1 6 i 6 n,

corresponding to the rows of the characteristic matrix Λ.

Lemma 7.3.26. The elements t1, . . . , tn form a linear regular sequence (an lsop)in the ring Z[P ]. Conversely, any lsop (7.16) in the ring Z[P ] defines a combina-torial quasitoric pair (P,Λ).

Proof. Since Z[P ] is a Cohen–Macaulay ring (Corollary 3.3.13), any lsop is aregular sequence by Proposition A.3.12. So it is enough to show that t1, . . . , tn is anlsop. Condition (7.4) implies that for any vertex v = Fj1 ∩ · · · ∩Fjn the restrictionsof the elements t1, . . . , tn form a basis in the linear part of the polynomial ringZ[vj1 , . . . , vjn ]. By Lemma 3.3.1, this condition specifies lsop’s in the ring Z[P ].


Theorem 7.3.27 ([90]). Let M = M(P,Λ) be a quasitoric manifold with Λ =(λij), 1 6 i 6 n, 1 6 j 6 m. The cohomology of M is given by

H∗(M) = Z[v1, . . . , vm]/I,

where vi ∈ H2(M) is the class dual to the characteristic submanifold Mi, and I isthe ideal generated by elements of the following two types:

(a) vi1 · · · vik whenever Mi1 ∩ · · · ∩Mik = ∅ (the Stanley–Reisner relations);(b) the linear forms ti = λi1v1 + · · ·+ λimvm, 1 6 i 6 n.

In other words, H∗(M) is the quotient of the face ring Z[P ] by the ideal gen-erated by linear forms (7.16). We shall prove a more general result, covering bothTheorem 5.3.1 and Theorem 7.3.27, in the next section (see Theorem 7.4.35).

If the matrix Λ has refined form (7.6), then the linear relations between thecohomology classes can be written as

(7.17) vi = −λi,n+1vn+1 − · · · − λi,mvm, 1 6 i 6 n.

It follows that the classes vn+1, . . . , vm multiplicatively generate the ring H∗(M).If the cohomology ring of a manifold is not generated by two-dimensional

classes, then the manifold does not support a torus action turning it into a qu-asitoric manifold. For example, complex Grassmanians (except complex projectivespaces) are not quasitoric.

Remark. The structure of the cohomology ring of M given in Theorem 7.3.27allows one to describe H∗(M) as a module over the Steenrod algebra. Furthermore,the ring E∗(M) can be described easily for any complex-oriented cohomology the-ory E∗. We shall give such a description for the complex cobordism ring Ω∗

U (M)and the action of the Landweber–Novikov algebra in Chapter 9.

The Chern classes of the stably complex structure on M (see Corollary 7.3.15)can be also described easily:

Theorem 7.3.28. Let (M, cT ) be a quasitoric manifold with the canonical sta-bly complex structure defined by an omniorientation. Then, in the notation ofTheorem 7.3.27, we have the following expression for the total Chern class:

c(M) = 1 + c1(M) + · · ·+ cn(M) = (1 + v1) · · · (1 + vm) ∈ H∗(M).

The homology class dual to ck(M) ∈ H2k(M) is represented by the sum of thesubmanifolds π−1(G) ⊂M corresponding to all (n− k)-dimensional faces G ⊂ P .

Proof. The first statement holds since the stably complex structure on M isdefined by the isomorphism with the complex bundle ρ1⊕· · ·⊕ρm, and c(ρi) = 1+vi,1 6 i 6 m. To prove the second statement, note that

ck(M) =∑

16j1<···<jk6m

vj1 · · · vjk ∈ H2k(M).

Here summands vj1 · · · vjk for which Fj1 ∩· · ·∩Fjk = ∅ are zero by Theorem 7.3.27,and the remaining summands are dual to the submanifolds π−1(G), where G =Fj1 ∩ · · · ∩ Fjk is a (n− k)-dimensional face.


Exercises.

7.3.29. For any 2- or 3-dimensional simple polytope P , there exists a quasitoricmanifold over P . (Hint: use the model M(P, λ) and the 4-colour Theorem, cf. [90].)

7.3.30. Let P be the dual of a 2-neighbourly (e.g., cyclic) simplicial polytopeof dimension n > 4 with m > 2n vertices. Then there is no quasitoric manifoldover P . (Hint: use the argument from Example 3.3.4.)

7.3.31. Let M = M(P,Λ) be a quasitoric manifold and ρi the complex linebundle over M defined by (7.7). Show that the restriction of ρi to the characteristicsubmanifold Mi is isomorphic to the normal bundle of Mi, and the restriction of ρito the complement M \Mi is trivial.

7.3.32. Calculate the signs of the fixed points for the two omniorientations on S2

in Example 7.3.16 and compare this with the sign calculation from Example D.6.3.

7.3.33. Let M be a quasitoric manifold over an n-polytope P and let ZPbe the corresponding moment-angle manifold. Show that the Borel constructionsET n ×Tn M and ETm ×Tm ZP are homotopy equivalent. Deduce that the equi-variant cohomology H∗

Tn(M) is isomorphic to the face ring Z[P ].

7.3.34. Let (P,Λ) be the quasitoric pair shown in Fig. 7.3. Show that thecorresponding quasitoric manifold M(P,Λ) admits a T 2-invariant almost complexstructure, but is not homeomorphic to a toric variety. (Hint: read Section 9.5.)

(−1,−1)

(0,1) (1,0)

(1,0) (0,1)

(−1,−1)

Figure 7.3. A quasitoric manifold over a hexagon.

7.3.35. The complex Grassmanian Grk(Cn) of k-planes in Cn with 2 6 k 6 n−2does not support a torus action turning it into a (quasi)toric manifold.

7.3.36. Let M be an omnioriented quasitoric manifold over a polytope P . Letx = Fi1 ∩ · · · ∩ Fin be a vertex of P written as an intersection of n facets; itcorresponds to fixed point x of M . Show that

⟨vi1vi2 · · · vin , [M ]

⟩= σ(x),

where vi ∈ H2(M) is the generator corresponding to Fi (see Theorem 7.3.27),[M ] ∈ H2n(M) is the fundamental homology class of the oriented manifold M , andσ(x) is the sign of x (see Lemma 7.3.18).

7.3.37. The top Chern number of M is given by 〈cn(M), [M ]〉 = ∑v∈P σ(v)

where the sum is taken over all vertices v of P . (Hint: use Theorem 7.3.28.)

7.3.38. Show that if M is a toric manifold, then c1(M) 6= 0. Given an exampleof an omnioriented quasitoric manifold M with c1(M) = 0.

7.4. LOCALLY STANDARD T -MANIFOLDS AND TORUS MANIFOLDS 263

7.4. Locally standard T -manifolds and torus manifolds

In this section we consider two closely related classes of half-dimensional torusactions on even-dimensional manifolds.

Definition 7.4.1. A locally standard T -manifold is a smooth connected closedorientable 2n-dimensional manifold M with a locally standard action of an n-dimensional torus T = T n.

The orbit space Q =M/T of a locally standard T -manifold is a manifold withcorners, but, unlike the case of quasitoric manifolds, it may fail to be a simplepolytope. For example, a free smooth T -action with dimM = 2dimT is locallystandard, but the quotient Q does not have faces at all. The richer is the com-binatorics of Q, the more information about the topology of M can be retrievedfrom this combinatorics. The easiest way to make the combinatorics of the orbitspace Q ‘rich enough’ is to require the existence of T -fixed points, or 0-faces of Q;then the local standardness condition would also imply the existence of faces of alldimensions between 0 and n. We therefore come to the following definition:

Definition 7.4.2. A torus manifold is a smooth connected closed orientable2n-dimensional manifold M with an effective smooth action of an n-torus T suchthat the fixed point set MT is nonempty.

Since M is connected, the T -action is effective and dimM = 2dimT , it followsthat the fixed point set MT is isolated. Indeed, the tangential T n-representation ata fixed point is faithful (Exercise B.3.8), which implies that this normal space hasdimension at least 2n. Furthermore, MT is finite because M is compact. So thelast condition in the definition above only excludes the possibility when the numberof fixed points is zero.

Torus manifolds were introduced and studied in the works [206], [153], [154]of Hattori and Masuda. Homological aspects of this study which we present herewere developed in [209] and [204].

The classes of locally standard T -manifolds and torus manifolds contain bothtoric and quasitoric manifolds in their intersection (see Section 7.6). Quasitoricmanifolds are examples with the most regular structure of the orbit space: a simplepolytope is contractible as a topological space, and the topology of the manifold isdescribed fully in terms of the combinatorics of faces. In general, the topology ofa locally standard T -manifold depends not only on the combinatorics of the orbitspace, but also on its topology. However, even orbit spaces with trivial topologymay have combinatorics more complicated than that of a polytope. Examples aresimplicial posets reviewed in Sections 2.8 and 3.5.

Even in the case of toric manifolds, we have encountered with combinatorialstructures more general than simple polytopes. As we have seen in Section 7.2,the orbit space Q = V/T of a nonsingular compact toric variety V is a manifoldwith corners in which all faces, including Q itself, are acyclic, and all nonemptyintersections of faces are connected. We refer to such a manifold with corners as ahomology polytope. It is a genuine polytope when the toric manifold is projective,but in general Q may be not combinatorially equivalent to a convex polytope (seethe discussion in Section 7.6). As a result, the class of quasitoric manifolds doesnot include all nonsingular compact toric varieties (toric manifolds), which is notvery convenient. At the same time, one could expect that all topological properties


characterising quasitoric manifolds also hold in a more general situation when theorbit space is a homology (rather than combinatorial) polytope. This is indeed thecase, as is seen by the results of this section.

The cohomology ring of a locally standard T -manifold M is generated by itsdegree-two elements if and only if the orbit space Q is a homology polytope (Theo-rem 7.4.41). In this case, the cohomology ring has the structure familiar from toricgeometry: H∗(M) is isomorphic to the quotient of the face ring of the orbit spaceQ by an ideal generated by certain linear forms.

More generally, we consider T -manifolds M whose cohomology vanishes in odddimensions. Such a T -manifold necessarily has a fixed point (Lemma 7.4.4), andis locally standard whenever dimM = 2dimT (Theorem 7.4.14). Therefore, underthe condition Hodd(M) = 0, the classes of locally standard T -manifolds and torusmanifolds coincide. In this case, the equivariant cohomology of M is a free finitelygenerated module over the T -equivariant cohomology of point, i.e. over the poly-nomial ring H∗(BT ) ∼= Z[t1, . . . , tn]. In other words, H∗

T (M) is a Cohen–Macaulayring. The orbit space of a T -manifold with vanishing odd-degree cohomology isnot necessarily a homology polytope, as is seen from a simple example of a half-dimensional torus action on an even-dimensional sphere (Example 7.4.11). There isa more general notion of a face-acyclic manifold with corners Q, in which all facesare acyclic, but the intersections of faces are not necessarily connected. It turnsout that the odd-degree cohomology of a T -manifold M vanishes if and only if theorbit space Q is face-acyclic (Theorem 7.4.46). The equivariant cohomology of suchM is isomorphic to the face ring of the simplicial poset of faces of Q (this ring mayno longer be generated in degree two, see Section 3.5).

The proofs use several results from the theory of GKM-manifolds and relatedGKM-graphs, whose foundations were laid in the work [129] of Goresky, Kottwitzand MacPherson. Relationship between this theory and torus manifolds is exploredfurther in Section 7.9.

Preliminaries: cohomology and fixed points. Here we obtain some pre-liminary results about torus actions, without assuming that the action is locallystandard. In this subsection M is a closed connected smooth orientable manifoldequipped with an effective smooth action of a torus T of arbitrary dimension.

We denote by MT the set of T -fixed points of M , which is a disjoint union offinitely many connected submanifolds.

Lemma 7.4.3. There exists a circle subgroup S ⊂ T such that MS =MT .

Proof. According to the standard result [41, Theorem IV.10.5], there is only afinite number of orbit types of the T -action on M , i.e. only finitely many subgroupsof T appear as the stabilisers of the action. We have MS =MT whenever S is notcontained in any proper stabiliser subgroup G ⊂ T . This condition is obviouslysatisfied for a generic circle S ⊂ T .

Lemma 7.4.4. If Hodd(M ;Q) = 0, then M has a T -fixed point.

Proof. Choose a generic circle S ⊂ T satisfyingMS =MT (see Lemma 7.4.3).If M does not have T -fixed points, then it also does not have S-fixed points,which implies that the Euler characteristic χ(M) is zero. On the other hand, ifHodd(M ;Q) = 0, then χ(M) > 0: a contradiction.


The inclusionMT →M induces the restriction map in equivariant cohomology:

(7.18) r : H∗T (M)→ H∗

T (MT ) = H∗(BT )⊗H∗(MT ).

The equivariant cohomology H∗T (M) is a H∗(BT )-module. We denote by H+(BT )

the positive-degree part of H∗(BT ). We shall need the following version of the‘localisation theorem’:

Theorem 7.4.5. The restriction map r : H∗T (M) → H∗

T (MT ) becomes an iso-

morphism when localised at H+(BT ).

For the proof, see [164, p. 40] or [144, Theorem 11.44].

Corollary 7.4.6. The kernel of the restriction map r : H∗T (M) → H∗

T (MT )

is a H∗(BT )-torsion module.

Since H∗(BT ) ∼= Z[t1, . . . , tn], we obtain that H∗T (M) is a free H∗(BT )-module

if and only if H∗T (M) is a Cohen–Macaulay ring (note that the H∗(BT )-module

H∗T (M) is finitely generated, as M is compact). The next statement gives a topo-

logical characterisation of T -manifolds with this property:

Lemma 7.4.7. Assume that MT is finite. Then H∗T (M) is a free H∗(BT )-

module if and only if Hodd(M) = 0. In this case, H∗T (M) ∼= H∗(BT )⊗H∗(M) as

H∗(BT )-modules.

Proof. Assume that Hodd(M) = 0. Then H∗(M) is torsion-free (an exercise)and the Serre spectral sequence of the bundle ρ : ET ×T M → BT collapses at E2.It follows that H∗

T (M) = H∗(ET ×T M) is isomorphic (as a H∗(BT )-module) tothe tensor product H∗(BT )⊗H∗(M), and therefore it is a free H∗(BT )-module.

Assume now that H∗T (M) is a free H∗(BT )-module. Consider the Eilenberg–

Moore spectral sequence of the bundle ρ : ET ×T M → BT (see Corollary B.4.4).It converges to H∗(M) and has

E∗,∗2 = Tor∗,∗H∗(BT )

(H∗T (M),Z

).

Since H∗T (M) is a free H∗(BT )-module,

Tor∗,∗H∗(BT )

(H∗T (M),Z

)= Tor0,∗H∗(BT )

(H∗T (M),Z

)= H∗

T (M)⊗H∗(BT ) Z

= H∗T (M)

/(ρ∗(H+(BT ))

).

Hence E0,∗2 = H∗

T (M)/(ρ∗(H+(BT ))) and E−p,∗

2 = 0 for p > 0. It follows that thespectral sequence collapses at E2, and

(7.19) H∗(M) = H∗T (M)

/(ρ∗(H+(BT ))

).

Since H∗T (M) is a free H∗(BT )-module, the restriction map (7.18) is a monomor-

phism (see Corollary 7.4.6). As MT is finite, H∗T (M

T ) is a sum of polynomial rings,hence Hodd

T (M) = 0. This together with (7.19) implies Hodd(M) = 0.

We shall be interested in the two special classes of T -manifolds M : those withvanishing odd-degree cohomology, and those with cohomology generated by thedegree-two classes. Both these cohomological properties are inherited by the fixedpoint set MH with respect to the action of any torus subgroup H ⊂ T . This factis proved in the next two lemmata; it enables us to use inductive arguments.

Lemma 7.4.8. Let H be a torus subgroup of T , and let N be a connected com-ponent of MH . If Hodd(M) = 0, then Hodd(N) = 0.


Proof. Choose a generic circle subgroup S ⊂ H with MS = MH , as inLemma 7.4.3. LetG ⊂ S be a subgroup of prime order p. The action ofG onH∗(M)is trivial, because G is contained in a connected group S. By [41, Theorem VII.2.2],dimHodd(MG;Zp) 6 dimHodd(M ;Zp). Hence Hodd(MG;Zp) = 0. Repeating thesame argument for the set MG with the induced action of the quotient group S/G(which is again a circle), we conclude that Hodd(MK ;Zp) = 0 for any p-subgroupK of S. However, MK = MS = MH if the order of K is sufficiently large, sowe obtain Hodd(MH ;Zp) = 0. Since p is an arbitrary prime, we conclude thatHodd(MH) = 0.

Lemma 7.4.9. Let M,H,N be as in Lemma 7.4.8. If the ring H∗(M) is gener-ated by its degree-two part, then the restriction map H∗(M)→ H∗(N) is surjective;in particular, the ring H∗(N) is also generated by its degree-two part.

Proof. It suffices to prove that the restriction map H∗(M ;Zp)→ H∗(N ;Zp)is surjective for any prime p, because Hodd(N) = 0 by Lemma 7.4.8.

The argument below is similar to that used in [41, Theorem VII.3.1]. As in theproof of Lemma 7.4.8, let S ⊂ H be a generic circle with MS =MH and let G ⊂ Sbe a subgroup of prime order p. By [41, Theorem VII.1.5], the restriction mapHkG(M ;Zp) → Hk

G(MG;Zp) is an isomorphism for sufficiently large k. Hence, for

any connected component N ′ of MG, the restriction r : HkG(M ;Zp)→ Hk

G(N′;Zp)

is surjective when k is large. Now consider the commutative diagram

H∗G(M ;Zp)

r−−−−→ H∗G(N

′;Zp) ∼= H∗(BG;Zp)⊗H∗(N ′;Zp)yy

H∗(M ;Zp)s−−−−→ H∗(N ′;Zp)

.

Choose a basis v1, . . . , vd ∈ H2(M ;Zp); then these elements multiplicatively gener-ate H∗(M ;Zp). Since Hodd(M ;Zp) = Hodd(MG;Zp) = 0 and χ(M) = χ(MT ) =χ(MG), it follows that

∑i dimHi(M ;Zp) =

∑i dimHi(MG;Zp). By [41, Theo-

rem VII.1.6], the Serre spectral sequence of the bundle EG×GM → BG collapsesat E2. Therefore, the vertical map H∗

G(M ;Zp)→ H∗(M ;Zp) in the above diagramis surjective. Let ξj ∈ H2

G(M ;Zp) be a lift of vj , and wj = s(vj). Let t be agenerator of H2(BG;Zp) ∼= Zp. We have H1(N ′;Zp) = 0 by Lemma 7.4.8, whichtogether with the commutativity of the diagram above implies r(ξj) = αjt+wj forsome αj ∈ Zp. Now let a ∈ H∗(N ′;Zp) be an arbitrary element. Then tℓa is in theimage of r when ℓ is large, i.e. there exists a polynomial P (ξ1, . . . , ξd) such that

r(P (ξ1, . . . , ξd)

)= tℓa.

On the other hand,

r(P (ξ1, . . . , ξd)

)= P (α1t+ w1, . . . , αdt+ wd) =

∑

k>0

tkQk(w1, . . . , wd)

for some polynomials Qk. Therefore, a = Qℓ(w1, . . . , wd), the restriction mapH∗(M ;Zp)→ H∗(N ′;Zp) is surjective, and H∗(N ′;Zp) is generated by the degree-two elements w1, . . . , wd.

Now we can repeat the same argument for N ′ with the induced action ofS/G, which is again a circle. We conclude that the restriction map H∗(M ;Zp) →H∗(N ′;Zp) is surjective for any p-subgroup K of S and any connected componentN ′ of MK . Now, if the order of K is sufficiently large, then MK = MS = MH


and hence N ′ = N . It follows that the restriction map H∗(M ;Zp)→ H∗(N ;Zp) issurjective for any connected component N of MH and for arbitrary prime p.

Characteristic submanifolds. From now on we assume that M is a torusmanifold, i.e. M is closed, connected, smooth and orientatble, dimM = 2dimT =2n, the T -action is smooth and effective and MT 6= ∅.

A closed connected codimension-two submanifold of M is called characteristicif it is fixed pointwise by a circle subgroup S ⊂ T . Since MT 6= ∅, there exists atleast one characteristic submanifold (an exercise). Furthermore, any fixed point iscontained in an intersection of some n characteristic submanifolds.

There are finitely many characteristic submanifolds in M , and we shall denotethem by Mi, 1 6 i 6 m. The intersection of any k 6 n characteristic submanifoldsis either empty or a (possibly disconnected) submanifold of codimension 2k fixedpointwise by a k-dimensional subtorus of T . In particular, the intersection of anyn characteristic submanifolds consists of finitely many T -fixed points.

Since M is orientable, each Mi is also orientable (since it is fixed by a circleaction). We say that M is omnioriented if an orientation is specified for M and foreach characteristic submanifold Mi. There are 2m+1 choices of omniorientationson M . In what follows we shall always assume M to be omnioriented. This allowsus to view the circle fixing Mi as an element in the integer lattice Hom(S, T ) ∼= Zn.

Since both M and Mi are oriented, the equivariant Gysin homomorphismH∗T (Mi) → H∗+2

T (M) is defined (see Section B.3). Denote by τi ∈ H2T (M) the

image of 1 ∈ H0T (Mi) under this homomorphism. The restriction of τi to H2

T (Mi)is the equivariant Euler class eT (νi) of the normal bundle νi = ν(Mi ⊂M).

Proposition 7.4.10. Let λi ∈ H2(BT ) be the element corresponding to thecircle subgroup fixing Mi via the identification H2(BT ) = Hom(S, T ).

(a) Let v be a T -fixed point of M , and assume that v is contained in theintersection Mi1 ∩· · ·∩Min . Then the corresponding elements λi1 , . . . , λinform a basis of H2(BT ) ∼= Zn. This basis is conjugate to the set of weightsw1(v), . . . ,wn(v) of the tangential T -representation at v.

(b) Let ρ : ET ×T M → BT be the projection. For any t ∈ H2(BT ),

ρ∗(t) =

m∑

i=1

〈t, λi〉τi modulo H∗(BT )-torsion.

Proof. Since the action is effective and M is connected, the tangential T -representation at v is faithful, so the set of weights w1(v), . . . ,wn(v) is a basis ofHom(T, S) = H2(BT ). The proof of (a) is the same as that of Proposition 7.3.17.

By Theorem 7.4.5, the restriction map (7.18) is an isomorphism after localisa-tion at H+(BT ). Therefore, to prove (b), we can apply the map r to the both sidesof the identity in question and verify the resulting identity in H∗(BT )⊗H∗(MT ).We may write r =

⊕v rv, where rv : H

∗T (M)→ H∗(BT ) is the map induced by the

inclusion of the fixed point v →M . We have rv(ρ∗(t)) = t and

rv

( m∑

i=1

〈t, λi〉τi)=

n∑

k=1

〈t, λik〉eT (νik)|v =

n∑

k=1

〈t, λik 〉wk(v) = t,

where the last identity holds because λi1 , . . . , λin and w1(v), . . . ,wn(v) are conju-gate bases.


Here is an example of a locally standard torus manifold, which is not quasitoric:

Example 7.4.11. Consider the unit 2n-sphere in Cn × R:

S2n =(z1, . . . , zn, y) ∈ Cn × R : |z1|2 + · · ·+ |zn|2 + y2 = 1

.

Define a T -action by the formula

(t1, . . . , tn) · (z1, . . . , zn, y) = (t1z1, . . . , tnzn, y).

There are two fixed points (0, . . . , 0,±1) and n characteristic submanifolds givenby z1 = 0, . . . , zn = 0. The intersection of any k characteristic submanifolds isconnected if k 6 n− 1, and consists of two disjoint fixed points if k = n > 1.

Unlike quasitoric manifolds, some intersections of characteristic submanifoldsare disconnected in the example above. The cohomology ring a quasitoric man-ifold is generated in degree two (Theorem 7.3.27). The next lemma shows thatthis is exactly the condition that guarantees the connectedness of intersections ofcharacteristic submanifolds.

Lemma 7.4.12. Suppose that H∗(M) is generated in degree two. Then allnonempty multiple intersections of characteristic submanifolds are connected andhave cohomology generated in degree two.

Proof. By Lemma 7.4.9, the cohomology H∗(Mi) is generated by the degree-two part and the restriction map H∗(M)→ H∗(Mi) is surjective for any character-istic submanifold Mi. Then it follows from Lemma 7.4.7 that the restriction mapH∗T (M)→ H∗

T (Mi) in equivariant cohomology is also surjective.Now we prove that multiple intersections are connected. Suppose that Mi1 ∩

· · · ∩ Mik 6= ∅, 1 < k 6 n, and let N be a connected component of this in-tersection. Since N is fixed by a subtorus, it contains a T -fixed point by Lem-mata 7.4.4 and 7.4.8. For each i ∈ i1, . . . , ik, there are embeddings ϕi : N →Mi,ψi : Mi →M , and the corresponding equivariant Gysin homomorphisms:

H0T (N)

ϕi!−−−−→ H2k−2T (Mi)

ψi!−−−−→ H2kT (M).

The map ψ∗i : H

∗T (M) → H∗

T (Mi) is surjective, so we obtain ϕi!(1) = ψ∗i (u) for

some u ∈ H2k−2T (M). Now we calculate

(ψi ϕi)!(1) = ψi!(ϕi!(1)) = ψi!(ψ∗i (u)

)= ψi!(1)u = τiu.

Hence (ψi ϕi)!(1) is divisible by τi, for each i ∈ i1, . . . , ik. By [206, Propo-sition 3.4] (see also Theorem 7.4.33 below), the degree-2k part of H∗

T (M) is ad-

ditively generated by monomials τk1j1 · · · τkpjp

such that Mj1 ∩ · · · ∩Mjp 6= ∅ and

k1 + · · · + kp = k. It follows that (ψi ϕi)!(1) is a nonzero integral multiple ofτi1 · · · τik ∈ H2k

T (M). By the definition of Gysin homomorphism, (ψi ϕi)!(1) mapsto zero under the restriction map H∗

T (M)→ H∗T (x) for any point x ∈ (M\N)T . On

the other hand, the image of τi1 . . . τik under the map H∗T (M)→ H∗

T (x) is nonzerofor any T -fixed point x ∈Mi1∩· · ·∩Mik . Thus, N is the only connected componentof the latter intersection. The fact that H∗(N) is generated by its degree-two partfollows from Lemma 7.4.9.


Orbit quotients and manifolds with corners. Let Q =M/T be the orbitspace of a locally standard T -manifold M , and let π : M → Q be the quotientprojection. Then Q is a manifold with corners (see Definition 7.1.3). The facets ofQ are the projections of the characteristic submanifolds: Fi = π(Mi), 1 6 i 6 m.Faces of Q are connected components of intersections of facets (note that the theseintersections may be disconnected, as in Example 7.4.11). For convenience, weregard Q itself as a face; all other faces are called proper.

If Hodd(M) = 0, then each face has a vertex by Lemmata 7.4.8 and 7.4.4.Moreover, if H∗(M) is generated in degree two, then all intersections of facets areconnected by Lemma 7.4.12.

Proposition 7.4.13. The orbit space Q of a locally standard T -manifold is anice manifold with corners.

Proof. We need to show that any face G of codimension k in Q is an intersec-tion of exactly k facets. Let q be a point in the interior of G, and let x ∈ π−1(q). Letz1, . . . , zn be the coordinates in a locally standard chart containing x. The pointx has exactly k of these coordinates vanishing, assume that these are z1, . . . , zk.Then, for any i = 1, . . . , k, the equation zi = 0 specifies the tangent space at x to acharacteristic submanifold of M . Therefore, x is contained in exactly k character-istic submanifolds. Each characteristic submanifold is projected onto a facet of Q,so that G is contained in exactly k facets.

Theorem 7.4.14. A torus manifold M with Hodd(M) = 0 is locally standard.

Proof. We first show that there are no nontrivial finite stabilisers for the T -action on M . Assume the opposite, i.e. there is a point x ∈M with finite nontrivialstabiliser Tx. Then Tx contains a nontrivial cyclic subgroup G of prime order p.Let N be the connected component of MG containing x. Since N contains x andTx is finite, the principal (i.e. the smallest) stabiliser of the induced T -action onN is finite. As in the proof of Lemma 7.4.8, it follows from [41, Theorem VII.2.2]that Hodd(N ;Zp) = 0. In particular, the Euler characteristic of N is non-zero,hence N has a T -fixed point, say y. The tangential T -representation TyM at y isfaithful, dimM = 2dimT and TyN is a proper T -subrepresentation of TyM . Itfollows that there is a nontrivial subtorus T ′ ⊂ T which fixes TyN and does not fixthe complement of TyN in TyM . Then T ′ is the principal stabiliser of N , whichcontradicts the above observation that the principal stabiliser of N is finite.

If the stabiliser Tx is trivial, M is obviously locally standard near x. Supposethat Tx is non-trivial. Then it cannot be finite, i.e. dim Tx > 0. Let H be theidentity component of Tx, and N the connected component of MH containing x.By Lemmata 7.4.8 and 7.4.4, N has a T -fixed point, say y. Looking at the tangentialrepresentation at y, we observe that the induced action of T/H on N is effective.By the previous argument, no point of N has a nontrivial finite stabiliser for theinduced action of T/H , which implies that Tx = H . Now x and y are both in thesame connected submanifold N fixed pointwise by Tx, hence the Tx-representationTxM agrees with the restriction of the tangential T -representation TyM to Tx. Thisimplies that M is locally standard near x.

Recall that a space X is acyclic if Hi(X) = 0 for any i.


Definition 7.4.15. We say that a manifold with corners Q is face-acyclic if Qand all its faces are acyclic. We call Q a homology polytope if it is face-acyclic andall nonempty multiple intersections of facets are connected.

A simple polytope is a homology polytope. Here is an example that does notarise in this way:

Example 7.4.16. The torus manifold S2n with the T -action from Exam-ple 7.4.11 is locally standard, and the map

(z1, . . . , zn, y)→ (|z1|, . . . , |zn|, y)induces a face preserving homeomorphism from the orbit space S2n/T to the space

(x1, . . . , xn, y) ∈ Rn+1 : x21 + · · ·+ x2n + y2 = 1, x1 > 0, . . . , xn > 0.This manifold with corners is face-acyclic, but is not a homology polytope if n > 1.

Proposition 7.4.10 allows us to define a characteristic map for locally standardtorus manifolds:

(7.20)λ : F1, . . . , Fm → H2(BT ) = Hom(S, T ) ∼= Zn,

Fi 7→ λi.

Given a point q ∈ Q, consider the smallest faceG(q) containing q. This face is a con-nected component of an intersection of facets Fi1 ∩· · ·∩Fik . We define the subtorusT (q) ⊂ T generated by the circle subgroups corresponding to λ(Fi1 ), . . . , λ(Fik ),and the identification space

(7.21) M(Q, λ) = Q× T/∼ where (x, t1) ∼ (x, t2) if t−11 t2 ∈ T (q).

It is easy to see that M(Q, λ) is a closed manifold with a T -action. Here is astraightforward generalisation of [90, Lemma 1.4] (see Proposition 7.3.6):

Proposition 7.4.17. Let M be a locally standard torus manifold with orbitspace Q, and let λ be the map defined by (7.20). If Q is face-acyclic, then there isa weakly T -equivariant homeomorphism

M(Q, λ)→M

covering the identity on Q.

Remark. Instead of face-acyclicity, one can only require that the second co-homology group of each face of Q vanishes, as this is the condition implying thetriviality of torus principal bundles obtained after blowing up the singular strataof lower dimension.

Face rings of manifolds with corners. Let Q be a nice manifold withcorners. The set of faces of Q containing a given face is isomorphic to the posetof faces of a simplex. In other words, the faces of Q form a simplicial poset S (see

Definition 2.8.1) with respect to the reverse inclusion. The initial element 0 of thisposet is Q. We refer to this simplicial poset S as the dual of Q; this duality extendsthe combinatorial duality between simple polytopes and their boundary spheretriangulations. The dual poset S is the poset of faces of a simplicial complex K ifand only if all nonempty multiple intersections of facets of Q are connected. In thiscase, K is the nerve of the covering of ∂Q by facets.


Example 7.4.18. Consider the three structures of a manifold with corners on adisc D2, shown in Fig. 7.4. The manifold with corners shown on the left is not nice.The middle one is nice and is face-acyclic, but is not a homology polytope. Theright one is homeomorphic to a 2-simplex, so it is a homology polytope. Comparethis with simplicial posets from Example 2.8.2.

s

(1)

ss

(2)

sss

(3)

Figure 7.4. 2-disc as a manifold with corners.

Example 7.4.19. Let Q = S2n/T be the orbit space of the torus manifold S2n,see Examples 7.4.11 and 7.4.16 (the case n = 2 is shown in Fig. 7.4 (2)). Here wehave n facets, the intersection of any k facets is connected if k 6 n − 1, but theintersection of n facets consists of two points. The dual simplicial cell complex isobtained by gluing two (n− 1)-simplices along their boundaries.

We can define the face ring of the orbit space Q as the face ring of the dualsimplicial poset (see Definition 3.5.2). However, for reader’s convenience, we givethe definition and state the main properties directly in terms of the combinatoricsof faces of Q. The proofs of the statements in this subsection are obtained byobvious dualisation of the corresponding statements in Section 3.5.

The intersection of two faces G,H in a manifold with corners can be discon-nected. We considerG∩H as a set of its connected components and use the notationE ∈ G ∩ H for connected components E of this intersection. If G ∩H 6= ∅, thenthere exists a unique minimal face G ∨H containing both G and H .

Definition 7.4.20. The face ring of a nice manifold with corners Q is thequotient

Z[Q] = Z[vG: G a face]/IQ,

where IQ is the ideal generated by vQ− 1 and all elements of the form

vGvH− v

G∨H·∑

E∈G∩H

vE.

In particular, if G ∩H = ∅, then vGvH= 0 in Z[Q].

The grading is given by deg vG= 2 codimG.

Example 7.4.21. Consider the torus action on S2n from Examples 7.4.11and 7.4.16 and let n = 2. Then Q is a 2-disc with two 0-faces, say p and q,and two 1-faces, say G and H . Then

Z[Q] = Z[vG, v

H, vp, vq]/(vGvH = vp + vq, vpvq = 0),

where deg vG= deg v

H= 2, deg vp = deg vq = 4. This is the same ring as the one

described in Example 3.5.3.1, but written in the dual notation.

Here is a dualisation of Theorem 3.5.7:


Theorem 7.4.22. Any element a ∈ Z[Q] can be written uniquely as an integrallinear combination of monomials vi1

G1vi2G2· · · vin

Gncorresponding to chains of faces

G1 ⊃ G2 ⊃ · · · ⊃ Gn of Q.

Given a vertex v ∈ Q, define the restriction map

sv : Z[Q]→ Z[Q]/(vG: G 6∋ v).

The ring Z[Q]/(vG: G 6∋ v) is identified with the polynomial ring Z[v

Fi1, . . . , v

Fin]

on n degree-two generators corresponding to the facets Fi1 , . . . , Fin containing v.

Theorem 7.4.23. Assume that each face of Q has a vertex. Then the sums =

⊕v sv of the restriction maps over all vertices v ∈ Q is a monomorphism from

Z[Q] to a direct sum of polynomial rings.

Equivariant cohomology. Here we construct a natural homomorphism fromthe face ring Z[Q] to the equivariant cohomology ring H∗

T (M) of a locally standardtorus manifold modulo H∗(BT )-torsion. Then we obtain conditions under whichthis homomorphism is monic and epic; in particular, we show that Z[Q]→ H∗

T (M)is an isomorphism when Hodd(M) = 0.

Since the fixed point set MT is finite, the restriction map (7.18) defines a map

(7.22) r =⊕

v∈MT

rv : H∗T (M)→ H∗

T (MT ) =

⊕

v∈MT

H∗(BT )

to a direct sum of polynomial rings. Its kernel is a H∗(BT )-torsion by Corol-lary 7.4.6, and r is a monomorphism when Hodd(M) = 0.

The 1-skeleton of Q is an n-valent graph. We identify MT with the verticesof Q and denote by E(Q) the set of oriented edges. Given an element e ∈ E(Q),denote the initial point and the terminal point of e by i(e) and t(e), respectively.Then Me = π−1(e) is a 2-sphere fixed pointwise by a codimension-one subtorus inT , and it contains two T -fixed points i(e) and t(e). The 2-dimensional subspaceTi(e)Me ⊂ Ti(e)M is an irreducible component of the tangential T -representationTi(e)M . The same is true for the other point t(e), and the T -representations Ti(e)Mand Tt(e)M are isomorphic. There is a unique characteristic submanifold, say Mi,intersecting Me at i(e) transversely. The omniorientation defines an orientation forthe normal bundle νi of Mi and, therefore, an orientation of Ti(e)Me. We therefore

can view the representation Ti(e)Me as an element of Hom(T, S) = H2(BT ), anddenote this element by α(e).

Let eT (νi) ∈ H2T (Mi) be the Euler class of the normal bundle, and denote its

restriction to a fixed point v ∈MTi by eT (νi)|v ∈ H2

T (v) = H2(BT ). Then

(7.23) eT (νi)|v = α(e),

where e is the unique edge such that i(e) = v and e /∈ Fi = π(Mi). Using theterminology of [146], we refer to the map

α : E(Q)→ H2(BT ), e 7→ α(e),

as an axial function.

Lemma 7.4.24. The axial function α has the following properties:

(a) α(e) = ±α(e) for any e ∈ E(Q), where e denotes e with the oppositeorientation;

(b) for any vertex v, the set αv = α(e) : i(e) = v is a basis of H2(BT );


(c) for e ∈ E(Q), we have αi(e) = αt(e) mod α(e).

Proof. Property (a) follows from the fact that Ti(e)Me and Tt(e)Me are iso-morphic as real T -representations, and (b) holds since the T -representation Ti(e)Mis faithful. Let Te be the codimension-one subtorus fixing Me. Then the Te-representations Ti(e)M and Tt(e)M are isomorphic, since the points i(e) and t(e)

are contained in the same connected component Me of MTe . This implies (c).

Remark. The original definition of an axial function in [146] requires theproperty α(e) = −α(e), but we allow α(e) = α(e). For example, α(e) = α(e) forthe T 2-action on S4 from Example 7.4.11.

Lemma 7.4.25. Given η ∈ H∗T (M) and e ∈ E(Q), the difference ri(e)(η) −

rt(e)(η) is divisible by α(e).

Proof. Consider the commutative diagram of restrictions

H∗T (M) −−−−→ H∗

T (i(e))⊕H∗T (t(e)) = H∗(BT )⊕H∗(BT )

yy

H∗Te(Me) −−−−→ H∗

Te(i(e))⊕H∗

Te(t(e)) =H∗(BTe)⊕H∗(BTe)

Since H∗Te(Me) = H∗(BTe) ⊗ H∗(Me), the two components of the image of η in

H∗(BTe) ⊕ H∗(BTe) above coincide. Then it follows from the commutativity ofthe diagram that the restrictions of ri(e)(η) and rt(e)(η) to H∗(BTe) coincide. Nowthe result follows from the fact that the kernel of the restriction map H∗(BT ) →H∗(BTe) is the ideal generated by α(e).

The preimage MG = π−1(G) of a codimension-k face G ⊂ Q is a closed T -invariant submanifold of M . It is a connected component of an intersection ofk characteristic submanifolds. The omniorientation defines an orientation of MG

and the equivariant Gysin homomorphism H0T (MG) → H2k

T (M). Let τG

denotethe image of 1 under this homomorphism; it is called the Thom class of MG. Therestriction of τ

G∈ H2k

T (M) to H2kT (MG) is the equivariant Euler class of the normal

bundle ν(MG ⊂M), and rv(τG) = 0 for v /∈ (MG)T . It follows from (7.23) that

(7.24) rv(τG) =

∏i(e)=v, e6⊂G

α(e) if v ∈ (MG)T ;

0 otherwise.

Define the quotient ring

H∗T (M) = H∗

T (M)/H∗(BT )-torsion.

The restriction map r from (7.22) induces a monomorphism H∗T (M) → H∗

T (MT ),

which we continue to denote by r. In particular, τG= 0 in H∗

T (M) if MG has noT -fixed points. The next lemma shows that the face ring relations from Defini-

tion 7.4.20 hold in H∗T (M) with v

Greplaced by τ

G.

Lemma 7.4.26. For any two faces G and H of Q, the relation

τGτH= τ

G∨H·∑

E∈G∩H

τE

holds in H∗T (M).


Proof. Since the map r : H∗T (M) → H∗

T (MT ) is injective, it suffices to show

that rv maps both sides of the identity to the same element, for any v ∈MT .Let v ∈MT . Given a face G ∋ v, set

Nv(G) = e ∈ E(Q) : i(e) = v, e 6⊂ G,which may be thought of as the set of directions transverse to G at v. Then we canrewrite (7.24) as follows:

(7.25) rv(τG) =∏

e∈Nv(G)

α(e),

where the right hand side is understood to be 1 if Nv(G) = ∅ and to be 0 if v /∈ G.Assume that v /∈ G∩H ; then either v /∈ G or v /∈ H , and v /∈ E for any E ∈ G∩H .Hence both sides of the identity from the lemma map to zero by rv. Assume thatv ∈ G ∩H ; then

Nv(G) ∪Nv(H) = Nv(G ∨H) ∪Nv(E),

where E is the connected component of the intersection G ∩ H containing v, andv /∈ E′ for any other E′ ∈ G∩H . This together with (7.25) implies that both sidesof the identity map to the same element by rv.

Lemma 7.4.26 implies that the map

Z[vG: G a face]→ H∗

T (M),

vG7→ τ

G

induces a homomorphism

(7.26) ϕ : Z[Q]→ H∗T (M).

Lemma 7.4.27. The homomorphism ϕ is injective if any face of Q has a vertex.

Proof. We have s = r ϕ, where s is the algebraic restriction map fromLemma 7.4.23. Since s is injective, ϕ is also injective.

The next theorem, which is a particular case of one of the main results of [129],says that when Hodd(M) = 0, the condition from Lemma 7.4.25 specifies preciselythe image of the equivariant cohomology under the restriction map:

Theorem 7.4.28 ([129], see also [144, Chapter 11]). Let M be a torus manifoldwith Hodd(M) = 0. Assume given an element ηv ∈ H∗(BT ) for each v ∈MT . Thenηv ∈

⊕v∈MT H∗(BT ) belongs to the image of the restriction map r in (7.22) if

and only if ηi(e) − ηt(e) is divisible by α(e) for any e ∈ E(Q).

Corollary 7.4.29. If Hodd(M) = 0, then the 1-skeleton of any face of Q(including Q itself) is connected.

Proof. By Theorem 7.4.28, ηv ∈⊕

v∈MT H0(BT ) belongs to r(H0T (M)) if

ηv is a locally constant function on the 1-skeleton of Q. On the other hand, sinceM is connected, the image r(H0

T (M)) is isomorphic to Z. Hence the 1-skeleton ofQ is connected. Similarly, the 1-skeleton of any face G of Q is connected, becauseMG = π−1(G) is also a torus manifold with Hodd(MG) = 0 (see Lemma 7.4.8).

Remark. Connectedness of 1-skeletons of faces of Q can be proven withoutreferring to Theorem 7.4.28, see the remark after Theorem 7.4.46.


For a face G ⊂ Q, we denote by I(G) the ideal in H∗(BT ) generated by allelements α(e) with e ∈ G.

Lemma 7.4.30. Suppose that the 1-skeleton of a face G is connected. Givenη ∈ H∗

T (M), if rv(η) /∈ I(G) for some vertex v ∈ G, then rw(η) /∈ I(G) for anyvertex w ∈ G.

Proof. Suppose rw(η) ∈ I(G) for some vertex w ∈ G. Then ru(η) ∈ I(G) forany vertex u ∈ G joined with w by an edge f ⊂ G, because rw(η)−ru(η) is divisibleby α(f) by Lemma 7.4.25. Since the 1-skeleton of G is connected, rw(η) ∈ I(G) forany vertex w ∈ G, which contradicts the assumption.

Lemma 7.4.31. If each face of Q has connected 1-skeleton, then H∗T (M) is

generated by the elements τG

as an H∗(BT )-module.

Proof. Let η ∈ H+T (M) be a nonzero element. Set

Z(η) = v ∈MT : rv(η) = 0.Take v /∈ Z(η). Then rv(η) ∈ H∗(BT ) is nonzero and we can express it as apolynomial in α(e) : i(e) = v, as the latter is a basis of H2(BT ). Let

(7.27)∏

i(e)=v

α(e)ne , ne > 0

be a monomial entering rv(η) with a nonzero coefficient. Let G be the face spannedby the edges e with ne = 0. Then rv(η) /∈ I(G) since rv(η) contains mono-mial (7.27). Hence rw(η) /∈ I(G) (in particular, rw(η) 6= 0) for any vertex w ∈ G,by Lemma 7.4.30.

On the other hand, it follows from (7.24) that monomial (7.27) can be writtenas rv(uGτG) with some u

G∈ H∗(BT ). Set η′ = η − u

GτG∈ H∗

T (M). We haverw(τG) = 0 for any w /∈ G, which implies rw(η

′) = rw(η) for w /∈ G. At thesame time, ru(η) 6= 0 for u ∈ G (see above). It follows that Z(η′) ⊃ Z(η). Thenumber of monomials in rv(η

′) is less than that in rv(η). Therefore, by subtractingfrom η a linear combination of elements τ

Gwith coefficients in H∗(BT ), we obtain

an element λ such that Z(λ) contains Z(η) as a proper subset. By iterating thisprocedure, we end up at an element whose restriction to every vertex is zero. Since

the restriction map r : H∗T (M)→ H∗

T (MT ) is injective, the result follows.

Theorem 7.4.32. Let M be a locally standard torus manifold with orbitspace Q. If each face of Q has connected 1-skeleton and contains a vertex, then

the monomorphism ϕ : Z[Q]→ H∗T (M) of (7.26) is an isomorphism.

Proof. The homomorphism ϕ is injective by Lemma 7.4.27. To prove that

ϕ is surjective it suffices to show that H∗T (M) is generated by the elements τ

Gas

a ring. By Proposition 7.4.10, the group H2T (M) is generated by the elements τ

Fi

corresponding to the facets Fi. (Note: the notation τi is used for τFi

in Propo-

sition 7.4.10.) In particular, any element in H2(BT ) ⊂ H∗T (M) can be written

as a linear combination of the elements τFi

. Hence any element in H∗(BT ) is apolynomial in τ

Fi. The rest follows from Lemma 7.4.31.

As a corollary we obtain a complete description of the equivariant cohomologyin the case Hodd(M) = 0:


Theorem 7.4.33 ([209, Corollary 7.6]). Let M be a locally standard T -manifoldwith Hodd(M) = 0. Then the equivariant cohomology H∗

T (M) is isomorphic to theface ring Z[Q] of the manifold with corners Q =M/T .

Proof. Indeed, if Hodd(M) = 0, then M is a torus manifold by Lemma 7.4.4.

Furthermore, H∗T (M) is a free H∗(BT )-module by Lemma 7.4.7, i.e. H∗

T (M) =H∗T (M). The result follows from Theorem 7.4.32.

The condition Hodd(M) = 0 can be interpreted in terms of the simplicial posetS dual to Q as follows:

Lemma 7.4.34. Let M be a torus manifold with quotient Q, and S be the faceposet of Q. Then Hodd(M) = 0 if and only if the following conditions are satisfied:

(a) the ring H∗T (M) is isomorphic to Z[S](= Z[Q]);

(b) Z[S] is a Cohen–Macaulay ring.

Furthermore, the ring H∗(M) is generated in degree two if and only if S is (theface poset of) a simplicial complex in addition to the above two conditions.

Proof. If Hodd(M) = 0, then H∗T (M) ∼= Z[Q] by Theorem 7.4.33, and Z[S] is

a Cohen–Macaulay ring by Lemma 7.4.7.Now we prove that Hodd(M) = 0 under conditions (a) and (b). The composite

H∗(BT )ρ∗−→ H∗

T (M)r−→

⊕

v∈MT

H∗(BT ).

is the diagonal map. By Lemma 3.5.8, this implies that ρ∗(t1), . . . , ρ∗(tn) is an

lsop. Since H∗T (M) is a Cohen–Macaulay ring, any lsop is a regular sequence

(Proposition A.3.12). It follows that H∗T (M) is a free H∗(BT )-module and hence

Hodd(M) = 0, by Lemma 7.4.7.It remains to prove the last statement. Assume that H∗(M) is generated in

degree two. By Lemma 7.4.12, all non-empty multiple intersections of facets areconnected. Then S is the nerve of the covering of ∂Q by facets.

Assume that S is a simplicial complex. Then Z[S] is generated in degree two.Furthermore, H∗

T (M) ∼= Z[S] is a free H∗(BT )-module by the first part of thetheorem, whence H∗(M) is a quotient ring of H∗

T (M). It follows that H∗(M) isalso generated by its degree-two part.

Ordinary cohomology. We can now describe the ordinary cohomology of alocally standard T -manifold (or torus manifold) with Hodd(M) = 0. This resultgeneralises the corresponding statements for toric and quasitoric manifolds (Theo-rems 5.3.1 and 7.3.27):

Theorem 7.4.35. Let M be a locally standard T -manifold with Hodd(M) = 0,and let Q =M/T be the orbit space. Then there is a ring isomorphism

H∗(M) ∼= Z[vG: G a face of Q]/I,

where I is the ideal generated by elements of the following two types:

(a) vGvH− v

G∨H

∑

E∈G∩H

vE;

(b)

m∑

i=1

〈t, λi〉vFi, for t ∈ H2(BT ).


Here Fi are the facets of Q, i = 1, . . . ,m, and the element λi ∈ H2(BT ) correspondsto the circle subgroup fixing the characteristic submanifold Mi = π−1(Fi).

The Betti numbers are given by the formula

rankH2i(M) = hi, 0 6 i 6 n,

where hi denote the components of the h-vector of the dual simplicial poset S.Proof. Since the Serre spectral sequence of the bundle ρ : ET ×T M → BT

collapses at E2, the map H∗T (M)→ H∗(M) is surjective and its kernel is the ideal

generated by all elements ρ∗(t), t ∈ H2(BT ). Therefore, the statement about thecohomology ring follows from Proposition 7.4.10 and Theorem 7.4.33.

By Lemma 7.4.7, H∗T (M) ∼= H∗(BT )⊗H∗(M) as H∗(BT )-modules, so we have

the following formula for the Poincare series:

F(H∗T (M);λ

)=

∑ni=0 rankH

2i(M)λ2i

(1− λ2)n .

On the other hand, the Poincare series of the face ring Z[Q] is given by Theo-rem 3.5.9, and the two series coincide by Theorem 7.4.33. This implies the state-ment about the Betti numbers.

Example 7.4.36. The equivariant cohomology ring of the torus manifold S4

from Example 7.4.21 is isomorphic to the ring Z[Q] described there. The ordinarycohomology ring is obtained by taking quotient by the ideal generated by v

Gand v

H.

Torus manifolds over homology polytopes. Using the previous results onthe equivariant and ordinary cohomology of torus manifolds with Hodd(M) = 0, wecan now proceed to describing the relationship between the cohomology of M andthe cohomology of its orbit space Q. Here we prove Theorem 7.4.41, which givesa cohomological characterisation of T -manifolds whose orbit spaces are homologypolytopes. In the next subsection we prove that Q is face-acyclic if and only ifHodd(M) = 0.

Lemma 7.4.37. If Hodd(M) = 0, then H1(Q) = 0.

Proof. We use the Leray spectral sequence of the projection map ET×TM →M/T = Q onto the second factor. It has Ep,q2 = Hp(M/T ;Hq) whereHq is the sheafwith stalk Hq(BTx) over a point x ∈ M/T , and the spectral sequence convergesto H∗

T (M). Since the T -action on M is locally standard, the isotropy group Tx atx ∈ M is a subtorus; so Hodd(BTx) = 0. Hence Hodd = 0, in particular, H1 = 0.

Moreover, H0 = Z (the constant sheaf). Therefore, we have E0,12 = 0 and E1,0

2 =H1(M/T ), whence H1(M/T ) ∼= H1

T (M). On the other hand, since Hodd(M) = 0by assumption, H∗

T (M) is a free H∗(BT )-module. Therefore, HoddT (M) = 0 by the

universal coefficient theorem. In particular, H1T (M) = 0.

Lemma 7.4.38. If either

(1) Q is a homology polytope, or(2) H∗(M) is generated by its degree-two part,

then the dual poset S of Q is (the face poset of) a Gorenstein* simplicial complex.

Proof. Under any of the assumptions (1) or (2), all nonempty multiple inter-sections of facets of Q are connected, so S is the face poset of the nerve simplicialcomplex K of the covering of ∂Q. For simplicity, we identify S with K.


We first prove that S is Gorenstein* under assumption (1). According to The-orem 3.4.2, it is enough to show that the link lk σ of any simplex σ ∈ K, hashomology of a sphere of dimension dim lk σ = n−2−dimσ. If σ = ∅ then lk σ is Kitself, and it has homology of an (n− 1)-sphere, since Q is a homology polytope. Ifσ 6= ∅ then lk σ is the nerve of a face of Q. Since any face of Q is again a homologypolytope, lk σ has homology of a sphere of dimension dim lkσ.

Now we prove that K is Gorenstein* under assumption (2). By Exercise 3.4.10,it is enough to show that

(a) K is Cohen–Macaulay;(b) every (n− 2)-dimensional simplex is contained in exactly two (n− 1)-di-

mensional simplices;(c) χ(K) = χ(Sn−1).

Condition (a) follows from Lemma 7.4.34. By definition, every k-dimensionalsimplex of K corresponds to a set of k + 1 characteristic submanifolds withnonempty intersection. By Lemma 7.4.12, the intersection of any n characteris-tic submanifolds is either empty or consists of a single T -fixed point. This meansthat (n − 1)-simplices of K are in one-to-one correspondence with T -fixed pointsof M . Now, each (n − 2)-simplex of K corresponds to a non-empty intersectionof n− 1 characteristic submanifolds of M . The latter intersection is connected byLemma 7.4.12, so it is a 2-sphere. Every 2-sphere contains exactly two T -fixedpoints, which implies (b). Finally, (c) is just the equation h0 = hn, which is validas hn = rankH2n(M) = 1.

Consider the order complex ord(S) (see Definition 2.3.6) and denote by C itsgeometric realisation. Then C is the cone over |S|. The space C has a face structure,as in the proof of Theorem 4.1.4. Namely, for each simplex σ ∈ ord(S) we denote byCσ the geometric realisation of the simplicial complex stσ = τ ∈ ord(S) : σ ⊂ τ.If σ has dimension (k− 1), then we say that Cσ is a codimension-k face of C. Eachfacet Ci (a face of codimension one) is the star of a vertex of ord(S), as in (4.4).A face of codimension k is a connected component of an intersection of k facets.Since any face is a cone, it is acyclic.

Although the face posets of C and Q coincide, the spaces themselves are dif-ferent: faces Cσ are defined abstractly and they are contractible (being cones), butfaces of Q may be not contractible even when Q is a homology polytope. Neverthe-less, we can define the characteristic map λ for the face structure of C by (7.20),and define a T -space

M(C, λ) = C × T/∼by analogy with (7.21). Since C may be not a manifold with corners, the spaceM(C, λ) is not a manifold in general. By a straightforward generalisation of Propo-sition 7.3.12, the space M(C, λ) can be identified with the quotient ZS/K of themoment-angle complex corresponding to S (see Section 4.10) by a freely actingtorus of dimension (m− n).

Proposition 7.4.39. We have H∗T (M(C, λ)) ∼= Z[S].

Proof. We have H∗Tm(ZS) ∼= Z[S] by Exercise 4.10.9. Now, H∗

Tm(ZS) ∼=H∗T (M(C, λ)), because M(C, λ) ∼= ZS/K and T = Tm/K.


Proposition 7.4.40. There is a face-preserving map Q→ C, which is coveredby a T -equivariant map

Φ: M(Q, λ)→M(C, λ).

Proof. The map Q→ C is constructed inductively; we start with a bijectionbetween vertices, and the extend the map to faces of higher dimension. Each faceof C is a cone, there are no obstructions to such extensions. Since the map Q→ Cpreserves the face structure, it is covered by a T -equivariant map

M(Q, λ) = T ×Q/∼ −→ T × C/∼=M(C, λ).

Now we can prove the main result of this subsection:

Theorem 7.4.41 ([209]). The cohomology of a locally standard T -manifold Mis generated in degree two if and only if the orbit space Q is a homology polytope.

Proof. Assume that Q is a homology polytope. Then M is homeomorphic tothe canonical model M(Q, λ) (Lemma 7.4.17), and we can view the map Φ fromProposition 7.4.40 as a map M → M(C, λ). For simplicity, we denote M(C, λ) byMC in this proof. Let MC,i = π−1(Ci), 1 6 i 6 m, be the ‘characteristic’ subspacesof MC . We also denote by ∂C the union of all facets Ci of C; topologically, ∂Cis the simplicial cell complex |S|. The T -action is free on MC\ ∪i MC,i and onM\ ∪iMi, so we have

H∗T (MC ,∪iMC,i) ∼= H∗(C, ∂C), H∗

T (M,∪iMi) ∼= H∗(Q, ∂Q).

Therefore, the map Φ induces a map between exact sequences

(7.28)

−→ H∗(C, ∂C) −−−−→ H∗T (MC) −−−−→ H∗

T (∪iMC,i) −→yyΦ∗

y

−→ H∗(Q, ∂Q) −−−−→ H∗T (M) −−−−→ H∗

T (∪iMi) −→Each Mi itself is a torus manifold with quotient homology polytope Fi. Us-ing induction and the Mayer–Vietoris sequence, we may assume that the mapH∗T (∪iMC,i) → H∗

T (∪iMi) above is an isomorphism. By Lemma 7.4.38, ∂C ∼= |S|has homology of an (n−1)-sphere. HenceH∗(C, ∂C) ∼= H∗(Dn, Sn−1), because C isthe cone over ∂C. We also haveH∗(Q, ∂Q) ∼= H∗(Dn, Sn−1), because Q is a homol-ogy polytope. Using these isomorphisms, we see from the construction of the mapΦ that the induced map H∗(C, ∂C)→ H∗(Q, ∂Q) is the identity on H∗(Dn, Sn−1).By applying the 5-lemma to (7.28) we obtain that Φ∗ : H∗

T (MC) → H∗T (M) is

an isomorphism. This together with Proposition 7.4.39 implies H∗T (M) ∼= Z[S].

The ring Z[S] is Cohen–Macaulay by Lemma 7.4.38. Therefore, all conditions ofLemma 7.4.34 are satisfied, and H∗(M) is generated by its degree-two part.

Assume now that that H∗(M) is generated in degree two. Since all nonemptyintersections of characteristic submanifolds are connected and their cohomologyrings are generated in degree two (Lemma 7.4.12), we may assume by inductionthat all proper faces of Q are homology polytopes. In particular, the proper facesare acyclic, whence H∗(∂Q) ∼= H∗(∂C), because both ∂Q and ∂C have acycliccoverings with the same nerve S. This together with Lemma 7.4.38 implies

(7.29) H∗(∂Q) ∼= H∗(Sn−1).

We need to show that Q itself is acyclic. We first prove the following:


Claim. H2(Q) = 0.

Proof. The claim is trivial for n = 1. If n = 2 then Q is a surface withboundary, hence H2(Q) = 0. Now assume n > 3. We consider the equivariantcohomology exact sequence of pair (M,∪iMi) (the bottom row of (7.28)). All themaps in the exact sequence are H∗(BT )-module maps. By Lemma 7.4.7, H∗

T (M) isa free H∗(BT )-module. On the other hand, H∗(Q, ∂Q) is finitely generated over Z,so it is a torsion H∗(BT )-module. It follows that the whole sequence splits intoshort exact sequences:

(7.30) 0→ HkT (M)→ Hk

T (∪iMi)→ Hk+1(Q, ∂Q)→ 0

By setting k = 1 we obtain

H1T (∪iMi) ∼= H2(Q, ∂Q).

The same argument as in Lemma 7.4.37 shows that

H1T (∪iMi) = H1((∪iMi)/T ) = H1(∂Q).

By considering the projection (ET ×M)/T → M/T = Q we conclude that thecoboundary map H1(∂Q)→ H2(Q, ∂Q) is an isomorphism. Therefore, we get thefollowing fragment of the exact sequence of pair:

0→ H2(Q)→ H2(∂Q)→ H3(Q, ∂Q).

By (7.29), H2(∂Q) ∼= H2(Sn−1), whence H2(Q) = 0 for n > 4. If n = 3, thecoboundary map H2(∂Q) → H3(Q, ∂Q) above is an isomorphism because Q isorientable by Lemma 7.4.37, whence H2(Q) = 0 again.

Now we resume the proof of the theorem. We obtain a T -homeomorphismM → M(Q, λ) (because H2(Q) = 0 and all proper faces are acyclic by the induc-tive assumption, see the remark after Proposition 7.4.17), and therefore a T -mapΦ: M → MC , as in the proof of the ‘if’ part of the theorem. We consider di-agram (7.28) again. Using induction and a Mayer–Vietoris argument, we mayassume that H∗

T (∪iMC,i) → H∗T (∪iMi) is an isomorphism. By Lemma 7.4.38,

H∗(C, ∂C) ∼= H∗(Dn, Sn−1). The map Q → C used in the construction of Φinduces a map

(7.31) H∗(Dn, Sn−1) ∼= H∗(C, ∂C)→ H∗(Q, ∂Q),

which is an isomorphism in dimension n. Applying the 5-lemma (Exercise A.1.3)to (7.28), we obtain that Φ∗ : H∗

T (MC) → H∗T (M) is injective. Theorem 7.4.33

and Proposition 7.4.39 imply that H∗T (M) ∼= Z[Q] ∼= H∗

T (MC), and all gradedcomponents of these rings are finitely generated. The same argument works withany field coefficients, so that Φ∗ : H∗

T (MC)→ H∗T (M) is actually an isomorphism.

By applying the 5-lemma again to diagram (7.28), we obtain that (7.31) is anisomorphism, i.e. H∗(Q, ∂Q) ∼= H∗(Dn, Sn−1). This together with (7.29) impliesthat Q is acyclic.

Theorem 7.4.41 shows that if the cohomology ring of a locally standard T -manifold is generated in degree two, then the combinatorics of the orbit space Qis fully determined by its nerve simplicial complex. The following result gives acharacterisation of simplicial complexes arising in this way.


Proposition 7.4.42. A simplicial complex K can be the nerve of a locallystandard T -manifold with cohomology generated in degree two if and only if K isGorenstein* and Z[K] admits an lsop.

Proof. If H∗(M) is generated in degree two, then K is Gorenstein* byLemma 7.4.38. In particular, Z[K] is a Cohen–Macaulay ring. Furthermore,H∗T (M) ∼= Z[K] by Theorem 7.4.33. Since H∗

T (M) ∼= H∗(BT ) ⊗ H∗(M) as aH∗(BT )-module, the ring Z[K] admits an lsop.

Now assume that Z[K] is Gorenstein* and admits an lsop. By [88, Theo-rem 12.2], there exists a homology polytope Q with nerve K. Since Z[K] admits anlsop, any element t ∈ H2(BT ) can be written as

t =

m∑

i=1

λi(t)vi

with λi(t) ∈ Z. Clearly, λi(t) is linear in t, so that we can view λi as an element ofthe dual lattice H2(BT ) (see Proposition 7.4.10). Now define a map λ (7.20) whichsends Fi to λi. Then M = M(Q, λ) (see (7.21)) is a locally standard T -manifold,and its cohomology is generated in degree two by Theorem 7.4.41.

Torus manifolds over face-acyclic manifolds with corners. Here weprove the second main result on the cohomology of T -manifolds, Theorem 7.4.46.It states that the orbit space Q of a locally standard T -manifold M is face-acyclicif and only if Hodd(M) = 0. The proof is by reduction to Theorem 7.4.41 on T -manifolds over homology polytopes; it relies upon the operation of blow-up and thealgebraic results of Section 3.6.

As before, M is a locally standard T -manifold with orbit projection π : M → Q.

Construction 7.4.43 (Blow-up of a T -manifold). Let MG = π−1(G) be thesubmanifold corresponding to a face G ⊂ Q, and ν

G= ν(MG ⊂ M) the normal

bundle. Since MG is a transverse intersection of characteristic submanifolds, νG

isthe Whitney sum of their normal bundles. The omniorientation on M makes ν

G

into a complex T -bundle.Consider the T -bundle ν

G⊕ C, where the T -action on the trivial summand C

is trivial. The projectivisation CP (νG⊕ C) is a locally standard T -manifold con-

taining MG, and there are invariant neighbourhoods of MG in M and of MG inCP (ν

G⊕C) which are T -diffeomorphic. After removing these invariant neighbour-

hoods of MG from M and CP (νG⊕ C) and reversing orientation on the latter,

we can identify the resulting T -manifolds along their boundaries. As a result, we

obtain a locally standard T -manifold M , which is called the blow-up of M at MG.

If G is a vertex, then M is diffeomorphic to the connected sum M # CPn.

There is a blow-down map M →M , which collapses the total space CP (νG⊕C)

onto MG and is the identity on the remaining part of M .

The orbit space Q of M is obtained by truncating Q at the face G. As a result,

Q acquires a new facet, which we denote by G. The simplicial cell complex dual to

Q is obtained from the dual of Q by applying a stellar subdivision at the face dualto G (see Definition 2.7.1).

Lemma 7.4.44. Q is face-acyclic if and only if Q is face-acyclic.

Proof. All new faces appearing as the result of truncating Q at G are con-

tained in the facet G ⊂ Q. The blow-down map M → M induces the projection


Q → Q collapsing G onto G. The face G is a deformation retract of G (combina-

torially, G is a product of G and a simplex). Hence G is acyclic if and only if G

is acyclic. Similarly, any other new face Q deformation retracts onto a face of Q.

Furthermore, the map Q→ Q is also a deformation retraction.

Lemma 7.4.45. Hodd(M) = 0 if and only if Hodd(M) = 0.

Proof. Assume that Hodd(M) = 0. By Lemma 7.4.8, Hodd(MG) = 0.The facial submanifold MG ⊂ M is blown up to a codimension-two submanifold

MG∼= CP (ν

G). The cohomology of the projectivisation of a complex vector bundle

over MG is a free H∗(MG)-module on even-dimensional generators (see, e.g. [297,

Chapter V]). Therefore, Hodd(MG) = 0.

The blow-down map M →M induces a map between exact sequences of pairs

Hk−1(MG) −−−−→ Hk(M,MG) −−−−→ Hk(M) −−−−→ Hk(MG)yy∼=

yy

Hk−1(MG) −−−−→ Hk(M, MG) −−−−→ Hk(M) −−−−→ Hk(MG)

where the second vertical arrow is an isomorphism by excision. Assume that k isodd. Since Hk(M) = 0, the map Hk−1(MG) → Hk(M,MG) is onto. Therefore,

Hk−1(MG)→ Hk(M, MG) is onto. Since Hk(MG) = 0, this implies Hk(M) = 0.

To prove the opposite statement, we use the algebraic results from Section 3.6.

Assume Hodd(M) = 0. Let S be the dual simplicial poset of Q, and let S be the

dual poset of Q. Then S is obtained from S by stellar subdivision at the face dual

to G. By Lemma 7.4.34, Z[S] is a Cohen–Macaulay ring. We claim that Z[S] is alsoCohen–Macaulay (i.e. the converse of Lemma 3.6.6 holds). Indeed, Theorem 3.6.7

implies that S is a Cohen–Macaulay simplicial poset. Let K be a simplicial complex

which is a common subdivision of simplicial cell complexes S and S (for example, we

may take K to be the barycentric subdivision of S). Then K is a Cohen–Macaulaycomplex by Corollary 3.6.2, hence S is a Cohen-Macaulay simplicial poset. Anotherapplication of Theorem 3.6.7 gives that Z[S] is a Cohen–Macaulay ring. Finally,Lemma 7.4.34 implies that Hodd(M) = 0.

Now we can prove our final result:

Theorem 7.4.46 ([209]). The odd-degree cohomology of a locally standard T -manifold M vanishes if and only if the orbit space Q is face-acyclic.

Proof. The idea is to reduce to Theorem 7.4.41 by blowing up sufficientlymany facial submanifolds. If the orbit space of M is face-acyclic, then it becomesa homology polytope after sufficiently many blow-ups.

Let S be the simplicial poset dual to Q. Since the barycentric subdivision is asequence of stellar subdivisions (Proposition 3.6.3), by applying appropriate blow-ups we get a torus manifold M ′ with orbit space Q′ such that the face poset of Q′

is the barycentric subdivision of the face poset of Q. The collapse map M ′ → Mis a composition of blow-down maps:

(7.32) M =M0 ←−M1 ←− · · · ←−Mk =M ′.

Assume that Hodd(M) = 0. Then M is locally standard by Theorem 7.4.14.By applying Lemma 7.4.45 successively, we get Hodd(M ′) = 0. By construction, all


intersections of faces of Q′ are connected, so H∗(M ′) is generated in degree two byLemma 7.4.34 and Q′ is a homology polytope by Theorem 7.4.41. In particular, Q′

is face-acyclic. Finally, by applying Lemma 7.4.44 successively, we conclude that Qis also face-acyclic.

Assume now that Q is face-acyclic. By applying Lemma 7.4.44 successively, weobtain that Q′ is also face-acyclic. On the other hand, S ′ is a simplicial complex,hence Q′ is a homology polytope. By Theorem 7.4.41, Hodd(M ′) = 0. By applyinglemma 7.4.45 successively, we finally conclude that Hodd(M) = 0.

Remark. As one can easily observe, the argument in the “only if” part of theabove theorem is independent of Theorem 7.4.28 and Theorem 7.4.33. Now, giventhat Q is face-acyclic, one readily deduces that the 1-skeleton of Q is connected.Indeed, otherwise the smallest face containing vertices from two different connectedcomponents of the 1-skeleton would be a manifold with at least two boundarycomponents and thereby non-acyclic. Thus, our reference to Theorem 7.4.28 wasactually irrelevant, although it made the argument more straightforward.

Exercises.

7.4.47. If M is orientable and Hodd(M) = 0, then H∗(M) is torsion-free.

7.4.48. Any torus manifold M has at least two characteristic submanifolds.

7.4.49. Any T -fixed point of a torus 2n-manifold M is contained in an inter-section of n characteristic submanifolds.

7.4.50. Each face of a face-acyclic manifold with corners has a vertex.

7.4.51. The equivariant Chern class of torus manifold M with an invariantstably complex structure is given by

cT (M) =

m∏

i=1

(1 + τi) modulo H∗(BT )-torsion

where τi ∈ H2T (M) is the Thom class defined before Proposition 7.4.10. (Hint: use

Corollary 7.4.6, see [206, Theorem 3.1].)

7.4.52. Let M be a torus manifold of dimension 2n with Hodd(M ;Z2) = 0and let G ⊂ T denote the discrete subgroup isomorphic to Zn2 . Show that theG-equivariant Stiefel–Whitney class of M is given by

wG(M) =

m∏

i=1

(1 + τi),

where τi ∈ H2G(M ;Z2) is the mod-2 Thom class.

7.4.53. Prove the following particular case of Theorem 3.7.4. Let S be a Goren-stein* simplicial poset such that there exists a torus manifold M with Hodd(M) = 0and orbit space Q whose dual poset is S. Let h(S) = (h0, h1, . . . , hn) be the h-vector. Assume that n is even and hi = 0 for some i. Then hn/2 is even. (Hint:use the previous exercise, see [209, Theorem 10.1]. An algebraic version of thisargument was used in [207] to prove Theorem 3.7.4 completely.)


7.5. Topological toric manifolds

Recall that a toric manifold is a smooth complete (compact) algebraic varietywith an effective algebraic action of an algebraic torus (C×)n having an open denseorbit. More constructively, toric varieties can be defined via the fan-variety corre-spondence; this gives a covering of the variety by invariant affine charts, which inthe smooth complete case are algebraic representation spaces of (C×)n.

The idea behind Ishida, Fukukawa and Masuda’s generalisation of toric mani-folds is to combine topological versions of these two definitions of toric varieties:

Definition 7.5.1 ([170]). A topological toric manifold is a closed smooth man-ifold X of dimension 2n with an effective smooth action of (C×)n having an opendense orbit and covered by finitely many invariant open subsets each equivariantlydiffeomorphic to a smooth representation space of (C×)n.

As is pointed out in [170], keeping only the first part of the definition (i.e. asmooth (C×)n-action with a dense orbit) leads to a vast and untractable class ofobjects; therefore it is important to include the covering by equivariant charts.

In this section we review the main properties of topological toric manifolds, andoutline the construction of the correspondence between topological toric manifoldsand generalised fans, called topological fans. We mainly follow the notation andterminology of [170]. The details of proofs can be found in the original paper.

The effectiveness of the (C×)n-action on a topological toric manifold X im-plies that the smooth representation of (C×)n modelling each invariant chart ofX is faithful. A faithful smooth real 2n-dimensional representation of (C×)n isisomorphic to a direct sum of complex one-dimensional representations.

To get a hold on smooth representations of (C×)n, we consider the case n = 1first. Since GL(1,C) = C×, a smooth representation of C× in C can be viewed asa smooth endomorphism of C×. Such an endomorphism has the form

z 7→ zµ = |z|b+ic( z|z|)a

with µ = (b+ ic, a) ∈ C× Z.

The representation given by z → zµ is algebraic if and only if c = 0 and b = a.Composition of smooth endomorphisms of C× defines a (noncommutative)

product on C× Z, given by

(zµ1)µ2 = zµ2µ1 , µ2µ1 = (b1b2 + i(b1c2 + c1a2), a1a2).

This product becomes the matrix product if we represent µ by 2× 2-matrices:(b2 0c2 a2

)(b1 0c1 a1

)=

(b2b1 0

c2b1 + a2c1 a2a1

).

Let R be the ring consisting of elements of C×Z with componentwise additionand multiplication defined above. The ring R is therefore isomorphic to the ringHomsm(C×,C×) of smooth endomorphisms of C×.

Given α = (α1, . . . , αn) ∈ Rn and β = (β1, . . . , βn) ∈ Rn, define smoothhomomorphisms χα ∈ Hom((C×)n,C×) and λβ ∈ Hom(C×, (C×)n) by

χα(z1, . . . , zn) =

n∏

k=1

zαk

k , λβ(z) = (zβ1

, . . . , zβn

),

7.5. TOPOLOGICAL TORIC MANIFOLDS 285

and also define

〈α, β〉 =n∑

k=1

αkβk ∈ R.

The following properties are checked easily:

(a) χα(λβ(z)) = z〈α,β〉;

(b) λβ(χα(z1, . . . , zn)) =

( n∏

k=1

zβ1αk

k , . . . ,

n∏

k=1

zβnαk

k

).

Given α1, . . . , αn ∈ Rn, define the endomorphism⊕n

i=1 χαi of (C×)n by

( n⊕

i=1

χαi

)(z1, . . . , zn) = (χα1(z1, . . . , zn), . . . , χ

αn(z1, . . . , zn)).

Proposition 7.5.2. Any smooth representation of (C×)n in Cn has the form⊕ni=1 χ

αi with αi ∈ Rn. This representation is faithful if and only if the n × n-matrix formed by the coordinates of αi has an inverse in Matn(R).

A faithful representation of (C×)n in Cn has a unique fixed point 0, hence the

fixed point set X(C×)n of a topological toric manifold is finite.A closed connected submanifold of real codimension two in a topological toric

manifold X is called characteristic if it is fixed pointwise by a subgroup isomorphicto C×. There are finitely many characteristic submanifolds in X , and we denotethem by X1, . . . , Xm.

It can be easily seen that a topological toric manifold X is simply connected([170, Proposition 3.2]). In particular, X is orientable. For each characteristicsubmanifold Xj , the normal bundle νj = ν(Xj ⊂ X) is orientable as a (C×)-equivariant bundle. Therefore,Xj itself is also orientable. A choice of an orientationfor each Xj together with an orientation of X is called an omniorientation on X .Topological toric manifolds are assumed to be omnioriented below.

Lemma 7.5.3 ([170, Lemma 3.3]). For each characteristic submanifold Xj,there is a unique βj(X) ∈ Rn such that the subgroup λβj(X)(C

×) ⊂ (C×)n fixes Xj

pointwise and λβj(X)(z)∗ξ = zξ for any z ∈ C×, ξ ∈ νi, where λβj(X)(z)∗ denotesthe differential of λβj(X)(z).

Characteristic submanifolds of X intersect transversely. Furthermore, multi-ple intersections of characteristic submanifolds are all connected (as in the case ofquasitoric manifolds, and unlike general torus manifolds), see [170, Lemma 3.6].

In particular, any fixed point v ⊂ X(C×)n is an intersection of an n-tupleXj1 , . . . , Xjn of characteristic submanifolds, so we have an isomorphism of real(C×)n-representation spaces

TvX ∼= (νj1 ⊕ · · · ⊕ νjn)|v.The omniorientation of X defines orientations for the left and right hand side of theidentity above. These two orientations may be different, so the sign of v is defined(compare Lemma 7.3.18 (a)).

The element βj(X) defined in Lemma 7.5.3 can be written as

(7.33) βj(X) = (bj(X) + icj(X),aj(X)) ∈ Cn × Zn.

Here is an analogue of Proposition 7.3.17 for topological toric manifolds:


Lemma 7.5.4 ([170, Lemma 3.4]). Let v = Xj1 ∩ · · · ∩ Xjn be a fixed pointof X. Then bj1(X), . . . , bjn(X) and aj1(X), . . . ,ajn(X) are bases of Rn andZn respectively.

The complex C×-representation space (νj1 ⊕ · · · ⊕ νjn)|v is isomorphic to⊕nk=1 χ

α(v)k , where α(v)

1 , . . . , α(v)n is the dual set of βj1(X), . . . , βjn(X), defined

uniquely by the condition ⟨α(v)k , βjℓ(X)

⟩= δkℓ

(here δkℓ denotes the Kronecker delta).

Define the simplicial complex K(X) on [m] whose simplices correspond tononempty intersections of characteristic submanifolds:

K(X) =I = i1, . . . , ik ∈ [m] : Xi1 ∩ · · · ∩Xik 6= ∅

.

When X is a toric manifold, we have bj(X) = aj(X) and cj(X) = 0 in (7.33),and the primitive vector aj(X) corresponds to the 1-parameter algebraic subgroupof (C×)n fixing the divisor Xj . Furthermore, the data K(X);a1(X), . . . ,am(X)define a complete simplicial regular fan (see Section 6.5). Now let us see what kindof combinatorial structure replaces a fan in the case of topological toric manifolds.

Given I ∈ K(X), let σI = R>〈bi : i ∈ I〉 denote the cone spanned by the vectorsbi ∈ Rn with i ∈ I. By Lemma 7.5.4, σI is a is a simplicial cone of dimension |I|.

Lemma 7.5.5 ([170, Lemma 3.7]).⋃I∈K(X) σI = Rn and σI ∩ σJ = σI∩J . In

other words, the data K(X); b1(X), . . . , bm(X) define a complete simplicial fan.

Definition 7.5.6. Let K be a simplicial complex on [m], and let

βj = (bj + icj ,aj) ∈ Cn × Zn, j = 1 . . . ,m

be a collection of m elements of Cn×Zn. The data K;β1, . . . , βm is said to definea (regular) topological fan ∆ if the following two conditions are satisfied:

(a) the data K; b1, . . . , bm define a simplicial fan in Rn;(b) for each I ∈ K, the set ai : i ∈ I is a part of basis of Zn.

A topological fan ∆ is said to be complete if the ordinary fan from (a) is complete.

Note that the fan of (a) is not required to be rational or regular, but if aj = bjfor all j, then ∆ becomes a regular ordinary fan.

Theorem 7.5.7 ([170, Theorem 8.1]). There is a bijective correspondence be-tween omnioriented topological toric manifolds of dimension 2n and complete topo-logical fans of dimension n.

Sketch of proof. Let X be a topological toric manifold. By Lemma 7.5.5,the data (K(X);β1(X), . . . , βm(X)) define a complete topological fan ∆(X).

Now let ∆ be a complete topological fan, defined by data (K;β1, . . . , βm). Foreach maximal simplex I = i1, . . . , in ∈ K, let αI1, . . . , αIn be the dual set ofβi1 , . . . , βin (compare Lemma 7.5.4). Condition (b) of Definition 7.5.6 guaran-

tees that the complex n-dimensional representation⊕n

k=1 χαI

k of (C×)n is faithful.These representation spaces corresponding to all maximal I ∈ K patch togetherinto a topological space X(∆) locally homeomorphic to Cn, and (C×)n acts on Xsmoothly with an open dense orbit. As in the case of ordinary fans, condition (b)of Definition 7.5.6 guarantees that the space X(∆) is Hausdorff, so it is a smoothmanifold. (However, the algebraic criterion for separatedness used in the proof of

7.5. TOPOLOGICAL TORIC MANIFOLDS 287

Lemma 5.1.4 cannot be used here; a topological argument is needed.) Finally thecondition that ∆ is complete gives that X(∆) is compact, i.e. closed.

An alternative way to proceed is to use an analogue of the quotient constructionof toric varieties, described in Section 5.4. To do this, define the coordinate subspacearrangement complement U(K) by (5.5), and define the homomorphism

λ : (C×)m → (C×)n, λ(z1, . . . , zm) =

m∏

k=1

λβk(zk).

Then λ is surjective and its kernel is given by

Kerλ =(z1, . . . , zm) ∈ (C×)m :

m∏

i=1

z〈α,βi〉i = 1 for any α ∈ Rn

,

by analogy with (5.3). Then define

X(∆) = U(K)/Kerλ =⋃

I∈K

(C,C×)I/Kerλ.

The space X(∆) has a smooth action of (C×)m/Kerλ ∼= (C×)n with an open denseorbit. Furthermore, for each maximal I ∈ K there is an equivariant diffeomorphism

ϕI : (C,C×)I/Kerλ→

n⊕

k=1

χαIk ,

where the latter is the faithful smooth (C×)n-representation space defined above.Condition (b) of Definition 7.5.6 translates into the condition of X(∆) being Haus-dorff. The fact that only the ‘real part’ (K; b1, . . . , bm) of the topological fan datamatters when deciding whether the quotient is Hausdorff should be clear from thesimilar argument in the proof of Theorem 6.5.2 (b). Finally, X(∆) is compact be-cause ∆ is complete. Thus, X(∆) with the local charts (C,C×)I/Kerλ, ϕI is atopological toric manifold.

For the classification of topological toric manifolds up to equivariant diffeomor-phism or homeomorphism, see [170, Corollary 8.2]

One can restrict the (C×)n-action on X to the compact n-torus Tn. Theresulting Tn-manifold X is obviously a locally standard torus manifold, so thequotient X/Tn is a manifold with corners.

Lemma 7.5.8 ([170, Lemma 7.1]). All faces of X/Tn are contractible and theface poset of X/Tn coincides with the inverse poset of K(X).

It follows that X/Tn is a homology polytope. As a corollary of Theorem 7.4.35and Theorem 7.4.41, we obtain the following description of the cohomology of X ,similar to toric or quasitoric manifolds:

Proposition 7.5.9. Let X be a topological toric manifold, whose associatedtopological fan is defined by the data (K(X);β1(X), . . . , βm(X)). Then the coho-mology ring of X is given by

H∗(X) ∼= Z[v1, . . . , vm]/I,where vi ∈ H2(X) is the class dual to the characteristic submanifold Xi, and I isthe ideal generated by elements of the following two types:

(a) vi1 · · · vik with i1, . . . , ik /∈ K(X);


(b)

m∑

i=1

⟨u,ai(X)

⟩vi, for any u ∈ Zn.

The element ai(X) ∈ Zn here is the second coordinate of βi(X), see (7.33).

7.6. Relationship between different classes of T -manifolds

The relationship is described schematically in Fig. 7.5. Each class shown in anoval is contained as a proper subclass in the next larger oval, except for one case(topological toric manifolds and quasitoric manifolds), where the relation is slightlymore subtle. Different examples are discussed below.

projectivetoric

manifolds

quasitoricmanifolds

toricmanifolds

topological toric manifolds

torus manifolds with Hodd = 0

locallystandard

T -manifolds

torusmanifolds

Figure 7.5. Classes of T -manifolds.

Projective toric manifolds are also Hamiltonian toric manifolds (see Sections 5.5and 6.3). However, when viewed as symplectic manifolds, projective toric manifoldsform a smaller class: their symplectic forms represent integral cohomology classesand their moment polytopes are lattice Delzant, while arbitrary Delzant polytopecan be realised as the moment polytope of a Hamiltonian toric manifold.

A toric manifold (nonsingular compact toric variety) which is not projective isdescribed in Example 5.2.3.

A projective toric manifold is quasitoric by Proposition 7.3.2.Many examples of quasitoric manifolds which are not toric can be constructed

using the equivariant connected sum operation (Construction 9.1.10). The simplestexample is CP 2 # CP 2. It can be easily seen to be a quasitoric manifold over a4-gon, but it does not admit an almost complex structure, and therefore cannot bea complex algebraic variety. A non-toric example with an invariant almost complexstructure is given in Exercise 7.3.34.

Examples of toric manifolds which is not quasitoric are constructed bySuyama [299]. The basic example is of real dimension 4; its corresponding regularsimplicial fan is obtained by subdividing a singular fan whose underlying simplicialcomplex is the Barnette sphere. More examples in arbitrary dimension can beconstructed by subsequent subdivision and suspension.

Any quasitoric manifold M is T -equivariantly homeomorphic to a T -manifoldobtained by restricting the (C×)n-action on a topological toric manifold to the com-pact torus Tn ⊂ (C×)n (see [170, Theorem 10.2]). The easiest way to see this is to

7.6. RELATIONSHIP BETWEEN DIFFERENT CLASSES OF T -MANIFOLDS 289

use the classification results (Proposition 7.3.11 and Theorem 7.5.7), and constructa topological fan from the combinatorial quasitoric pair (P,Λ) corresponding to M .To do this, consider any convex realisation (6.1) of the polytope P , and define

βj = (aj + icj , λj) ∈ Cn × Zn, j = 1, . . . ,m,

where aj are the normal vectors to the facets of P , and λj are the columns ofthe characteristic matrix Λ. The vectors cj ∈ Rn can be chosen arbitrarily. Thenthe data (KP ;β1, . . . , βm) define a topological fan ∆. Indeed, condition (a) fromDefinition 7.5.6 is satisfied because (KP ;a1, . . . ,am) define the normal fan ΣP , andcondition (b) is equivalent to (7.4). Then the topological toric manifold X(∆) isT -homeomorphic to M and the restriction of the (C×)n-action to the compact torusT gives the T -action on M (this follows by comparing the construction of X(∆)with Proposition 7.3.12).

Remark. One would expect that the T -action on a quasitoric manifold M canbe extended to a (C×)n-action which gives M a structure of a topologically toricmanifold. This stronger statement would hold if one can replace an equivarianthomeomorphism by an equivariant diffeomorphism in Proposition 7.3.6.

In [170, §9] there is constructed a topological toric manifoldX whose associatedsimplicial complex K(X) is the Barnette sphere (see Construction 2.5.5). Since theBarnette sphere is not polytopal, this X is not a quasitoric manifold.

A toric manifold is topologically toric by definition. An example of a topologicaltoric manifold which is not toric can be constructed from the quasitoric manifoldCP 2 # CP 2 as described above. An explicit topological toric atlas on CP 2 # CP 2

and the corresponding topological fan are described in [170, §5, Example].Any topological toric manifolds is a torus manifold with Hodd = 0 by Proposi-

tion 7.5.9. An even-dimensional sphere is a torus manifold with Hodd = 0 (Exam-ple 7.4.11), but it is not a topological toric manifold.

An example of a torus manifold with Hodd 6= 0 can be constructed as follows.Take any torus manifold M whose quotient Q is face-acyclic. Let R be any closed

manifold which is not a homology sphere. Consider the connected sum Q = Q#R

taken near an interior point of Q. Then Q is a manifold with corners with ∂Q =

∂Q. In particular, all proper faces of Q are acyclic, but Q itself is not (we have

H∗(Q) ∼= H∗(R \ pt)). Consider the manifold M =M(Q, λ) constructed using the

characteristic map of M , see (7.21). The singular T -orbits of M are the same as

those of M , but the free orbits are different. Now the quotient of M is Q, which

is not face-acyclic. Hence Hodd(M) 6= 0 by Theorem 7.4.46 (this can be also easilyseen directly). The simplest example is obtained whenM = CP 2 and R is a 2-torus.

Any torus manifold with Hodd = 0 is locally standard by Theorem 7.4.14.An example of a torus manifold which is not locally standard is given in [170,

§11]. A free action of T n on the first factor of a product manifold T n × Nn givesan example of a locally standard T -manifold which is not a torus manifold.

We conclude this section by mentioning that there are real analogues of allclasses of T -manifolds considered here, in which the torus Tn is replaced by the‘real torus’ (Z2)

n and the algebraic torus (C×)n is replaced by (R×)n, where R× ∼=R>×Z2 is the multiplicative group of real numbers. The ‘real’ versions of the resultsof this chapter are often simpler; the reader may recover the details of the proofshimself. Real toric varieties feature in tropical geometry [173]. Real quasitoric


manifolds are known as small covers of simple polytopes; they were introduced byDavis and Januszkiewicz in [90] along with quasitoric manifolds.

7.7. Bounded flag manifolds

Bounded flag manifolds BFn introduced by Buchstaber and Ray in [61] andsubsequently studied in [60] and [62]. Each BFn is a projective toric manifoldwhose moment polytope is combinatorially equivalent to an n-cube, so that BFn isalso a quasitoric manifold over a cube. Bounded flag manifolds are the examples ofiterated projective bundles, or Bott towers, which are studied in the next section.The manifolds BFn find numerous application in cobordism theory; they are im-plicitly present in the work of Conner–Floyd [81] and in the construction of Ray’sbasis [275] in complex bordism of CP∞ (see details in Section 9.2), as well as usedin the construction of toric representatives in complex bordism classes, described inSection 9.1. Bounded flag manifolds also illustrate nicely many constructions andresults related to toric and quasitoric manifolds.

Construction 7.7.1 (Bounded flag manifold). A bounded flag in Cn+1 is acomplete flag

U = U1 ⊂ U2 ⊂ · · · ⊂ Un+1 = Cn+1, dimUi = ifor which Uk, 2 6 k 6 n, contains the coordinate subspace Ck−1 = 〈e1, . . . , ek−1〉spanned by the first k − 1 standard basis vectors. Denote by BFn the set of allbounded flags in Cn+1.

Every bounded flag U in Cn+1 is uniquely determined by the set of n lines

(7.34) L = l1, . . . , ln : lk ⊂ Ck ⊕ lk+1 for 1 6 k 6 n, ln+1 = Cn+1,where Ck = 〈ek〉 is the kth coordinate line in Cn+1. Indeed, given a set of linesL as above, we can construct a bounded flag U by setting Uk = Ck−1 ⊕ lk for1 6 k 6 n+1. Conversely, the conditions lk ⊂ Ck⊕ lk+1 and Uk = Ck−1⊕ lk allowsus to reconstruct the set of lines L from a flag U in the reverse order ln+1, ln, . . . , l1.

Theorem 7.7.2. The action of the algebraic torus (C×)n on Cn+1 given by

(t1, . . . , tn) · (w1, . . . , wn, wn+1) = (t1w1, . . . , tnwn, wn+1),

where (t1, . . . , tn) ∈ (C×)n and (w1, . . . , wn, wn+1) ∈ Cn+1, induces an action onbounded flags, and therefore makes BFn into a smooth toric variety.

Proof. We first construct a covering of BFn by smooth affine charts withregular change of coordinates functions, thereby giving BFn a structure of a smoothaffine variety. We parametrise bounded flags by sets of lines (7.34). Let vk be anonzero vector in lk, for 1 6 k 6 n, and set vn+1 = en+1 for the last line. Considertwo collections of n opens subsets in BFn:

V 0k = U ∈ BFn : lk 6= Ck, V 1

k = U ∈ BFn : 〈vk, ek〉 6= 0, 1 6 k 6 n.

Now define 2n open subsets

V ε1,...,εn = V ε11 ∩ · · · ∩ V εnn , where εk = 0, 1,

Then V ε1,...,εn is a covering of BFn, because V 0k ∪ V 1

k = BFn for any k. Thecondition lk ⊂ Ck ⊕ lk+1 implies

(7.35) vk = zkek + zk+nvk+1, 1 6 k 6 n,

7.7. BOUNDED FLAG MANIFOLDS 291

for some zi ∈ C, 1 6 i 6 2n. We have zk 6= 0 if U ∈ V 1k , and zk+n 6= 0 if

U ∈ V 0k . Let U ∈ V ε1,...,εn ; then we can choose the vectors (7.35) in the form

vk = x0kek + vk+1 if εk = 0, and vk = ek + x1kvk+1 if εk = 1, for 1 6 k 6 n. Thenwe can identify V ε1,...,εn with Cn using the affine coordinates (xε11 , . . . , x

εnn ). The

change of coordinate functions are regular on intersections of charts by inspection,so that BFn is a smooth algebraic variety, with affine atlas V ε1,...,εn.

Furthermore, the change of coordinate functions are Laurent monomials, whichimplies that BFn is a toric variety. This can also be seen directly, as the torus actiondefined in the theorem is standard in the affine chart V 0,...,0, that is,

(t1, . . . , tn) · (x01, . . . , x0n) = (t1x01, . . . , tnx

0n).

Proposition 7.7.3. The complete fan Σ corresponding to the toric variety BFn

has 2n one-dimensional cones generated by the vectors

a0k = ek, a1

k = −e1 − · · · − ek, 1 6 k 6 n,

and 2n maximal cones generated by the sets of vectors aε11 , . . . ,aεnn , where εk = 0, 1.

Proof. Each affine chart V ε1,...,εn ⊂ BFn constructed in the proof of Theo-rem 7.7.2 corresponds to an n-dimensional cone σε1,...,εn of the fan Σ, so there are 2n

maximal cones in total. One-dimensional cones of Σ correspond to (C×)n-invariantsubmanifolds of complex codimension 1 in BFn. Each of these submanifolds isdefined by vanishing of one of the affine coordinates, i.e. by an equation xεkk = 0,so there are 2n such submanifolds.

In order to find the generators of the cone σε1,...,εn , we note that theprimitive generators of the dual cone (σε1,...,εn)v are the weights of the (C×)n-representation in the affine space Cn corresponding to the chart V ε1,...,εn . The(C×)n-representation in the chart V 0,...,0 is standard, so we have a0

k = ek for1 6 k 6 n. In order to find the remaining vectors it is enough to calculate theweights of the torus representation in the chart V 1,...,1.

The coordinates (x11, . . . , x1n) in the chart V 1,...,1 are defined from the relations

vk = ek + x1kvk+1, 1 6 k 6 n, and vn+1 = en+1. An element (t1, . . . , tn) ∈ (C×)n

acts on ek by multiplication by tk for 1 6 k 6 n and acts on en+1 identically (seeTheorem 7.7.2). Then it is easy to see that the torus representation is written inthe coordinates (x11, . . . , x

1n) as follows:

(t1, . . . , tn) · (x11, . . . , x1n) = (t−11 t2x

11, . . . , t

−1n−1tnx

1n−1, t

−1n x1n).

In other words, the weights of this representation are the columns of the matrix

W =

−1 0 . . . 0 01 −1 . . . 0 0...

.... . .

......

0 0 . . . −1 00 0 . . . 1 −1

.

The generators a11, . . . ,a

1n of the cone σ1,...,1 form the dual basis, i.e. they are

columns of the matrix (W−1)t. These are the vectors listed in the lemma.

Proposition 7.7.4. The bounded flag manifold BFn is the projective toricvariety corresponding to the polytope

P =x ∈ Rn : xk > 0, x1 + · · ·+ xk 6 k, for 1 6 k 6 n

.


Proof. We need to check that the normal fan of this P is the fan from Propo-sition 7.7.3. Indeed, the 2n inequalities specifying P can be written as 〈a0

k, x 〉 > 0and 〈a1

k, x 〉+ k > 0, for 1 6 k 6 n. Set

(7.36) F 0k = x ∈ P : 〈a0

k, x 〉 = 0 and F 1k = x ∈ P : 〈a1

k, x 〉+ k = 0.Then we need to check that

(a) each F εk (ε = 0, 1) is a facet of P ;(b) F 0

k ∩ F 1k = ∅, for 1 6 k 6 n;

(c) the intersection of any n-tuple F ε11 , . . . , F εnn is a vertex of P .

This is left as an exercise.

Proposition 7.7.5 ([62]). The bounded flag manifold BFn is a quasitoric man-ifold over a combinatorial n-cube In, with characteristic matrix

(7.37) Λ =

1 0 . . . 00 1 . . . 0...

.... . .

...0 0 . . . 1

∣∣∣∣∣∣∣∣∣

−1 −1 . . . −10 −1 . . . −1...

.... . .

...0 0 . . . −1

.

Proof. Indeed, the polytope from Proposition 7.7.4 is combinatorially equiv-alent to a cube.

Example 7.7.6. The manifold BF 2 is isomorphic to the Hirzebruch surface F1

(or F−1) from Example 5.1.8.

We can also describe BFn as a toric manifold using the quotient construction(Section 5.4) or symplectic reduction (Section 5.5), as follows. The moment-anglemanifold corresponding to a cube In is a product of n three-dimensional spheres:

ZIn = (z1, . . . , z2n) ∈ C2n : |zk|2 + |zk+n|2 = 1, 1 6 k 6 n.The manifold BFn is obtained by taking quotient of ZIn by the kernelK of the mapT2n → Tn given by matrix (7.37). We have K ∼= Tn, and the inclusion K(Λ) ⊂ T2n

is given by

(t1, . . . , tn) 7→ (t1t2· · · tn−1tn, t2· · · tn−1tn, . . . , tn−1tn, tn, t1, t2, . . . , tn).

Geometrically, the projection ZIn → BFn maps z = (z1, . . . , z2n) to the boundedflag defined by the set of lines l1, . . . , ln+1, where lk = 〈vk〉 and vk is given by (7.35)(an exercise).

The algebraic quotient description of BFn is very much similar. Instead of themoment-angle manifold ZIn ∼= (S3)n we have the space U(Σ) = (C2 \ 0)n. Themanifold BFn is obtained by taking quotient of U(Σ) by the kernel G of the mapof algebraic tori (C×)2n → (C×)n given by matrix (7.37).

Now we describe the characteristic submanifolds and their corresponding linebundles (7.7). Let π : BFn → P be the quotient projection for the torus action,and let ρεk denote the line bundle corresponding to the characteristic submanifold(or (C×)n-invariant divisor) π−1(F εk ), for 1 6 k 6 n, ε = 0, 1, see (7.36).

Proposition 7.7.7 ([62]).

(a) The characteristic submanifold π−1(F 0k ) is isomorphic to BFn−1, and

π−1(F 1k ) is isomorphic to BF k−1 × BFn−k, for 1 6 k 6 n.

7.8. BOTT TOWERS 293

(b) The line bundle ρ0k is isomorphic to the bundle whose fibre over U ∈ BFn

is the line lk = Uk/Ck−1. The line bundle ρ1k is isomorphic to the bundlewhose fibre over U ∈ BFn is the quotient (Ck ⊕ lk+1)/lk = Uk+1/Uk.

Proof. The submanifold π−1(F 0k ) ⊂ BFn is obtained by projecting the sub-

manifold of ZIn given by the equation zk = 0 onto BFn. If zk = 0, then the vectorsv1, . . . , vn defined by (7.35) all belong to the subspace C1,..., n+1\k. It followsthat π−1(F 0

k ) can be identified with the set of bounded flags in C1,..., n+1\k, thatis, with BFn−1.

Similarly, the submanifold π−1(F 1k ) is the projection of the submanifold of ZIn

given by the equation zk = 1. Then (7.35) implies that the vectors v1, . . . , vk belongto the subspace Ck ⊂ Cn+1, and the vectors vk+1, . . . , vn belong to the subspaceCk+1,..., n+1. Therefore, π−1(F 1

k ) can be identified with BF k−1 × BFn−k.Statement (b) also follows from (7.35), because the line bundle ρ0k is isomorphic

to ZIn ×K Ck, and ρ1k is isomorphic to ZIn ×K Cn+k.

Proposition 7.7.8. The manifold BFn is the complex projectivisation of thecomplex plane bundle C⊕ ρ01 over BFn−1.

Proof. Consider the projection BFn → BFn−1 taking a bounded flag U =U1 ⊂ U2 ⊂ · · · ⊂ Un ⊂ Cn+1 to the flag U ′ = U/C1 in C2,..., n+1 ∼= Cn. (Moreprecisely, U ′ = U ′

1 ⊂ U ′2 ⊂ · · · ⊂ U ′

n−1, where U ′k = Uk+1/C1, 1 6 k 6

n− 1.) The set of lines (7.34) corresponding to U ′ is obtained from the set of linescorresponding to U by forgetting the first line. In order to recover the flag U fromthe flag U ′, one needs to choose a line l1 in the plane C1 ⊕ l2. Since l2 is the firstline in the set corresponding to the flag U ′ ∈ BFn−1, we obtain BFn

∼= CP (C⊕ρ01),as needed.

We therefore obtain a tower of fibrations BFn → BFn−1 → · · · → BF 1 = CP 1,where each BF k is the projectivisation of a complex 2-plane bundle over BF k−1.In particular, the fibre of each bundle in the tower is CP 1. Towers of fibrationsarising in this way are called Bott towers ; they are the subject of the next section.

Exercises.

7.7.9. The fan described in Proposition 7.7.3 is the normal fan of the polytopefrom Proposition 7.7.4.

7.7.10. Given z = (z1, . . . , z2n), define the vectors vn+1 = en+1, vn, . . . , v1

by (7.35), and set lk = 〈vk〉, 1 6 k 6 n + 1. Then the projection ZIn → BFn

maps z ∈ ZIn to the bounded flag in Cn+1 defined by the set of lines l1, . . . , ln+1.

7.8. Bott towers

In their study of symmetric spaces, Bott and Samelson [37] introduced a fam-ily of complex manifolds obtained as the total spaces of iterated bundles over CP 1

with fibre CP 1. Grossberg and Karshon [137] showed that these manifolds carry analgebraic torus action, and therefore constitute an important family of smooth pro-jective toric varieties, and called them Bott towers. Civan and Ray [78] developedsignificantly the study of Bott towers by enumerating the invariant stably complexstructures and calculating their complex and real K-theory rings, and cobordism.


Each Bott tower is a projective toric manifold whose corresponding simplepolytope is combinatorially equivalent to a cube (a toric manifold over cube forshort). We have the following hierarchy of classes of T -manifolds:

Bott towers ⊂ toric manifolds over cubes ⊂ quasitoric manifolds over cubes

By the result of Dobrinskaya [96], the first inclusion above is in fact an identity(we explain this in Corollary 7.8.12).

Two results were obtained in [210] relating circle actions on Bott towers, theirtopological structure, and cohomology rings. First (Theorem 7.8.17), if a Bott toweradmits a semifree S1-action with isolated fixed points, then it is S1-equivariantlydiffeomorphic to a product of 2-spheres. Second (Theorem 7.8.23), a Bott towerwhose cohomology ring is isomorphic to that of a product of spheres is actuallydiffeomorphic to this product. Both theorems can be extended to quasitoric mani-folds over cubes, but only in the topological category (Theorems 7.8.19 and 7.8.25).These results have been further extended by several authors, and led to the study ofthe so-called cohomological rigidity property for different classes of manifolds withtorus actions; we discuss this circle of problems in the end of this section.

Definition and main properties.

Definition 7.8.1. A Bott tower of height n is a tower of fibre bundles

Bnpn−→ Bn−1

pn−1−→ · · · −→ B2p2−→ B1 −→ pt ,

of complex manifolds, where B1 = CP 1 and Bk = CP (C⊕ξk−1) for 2 6 k 6 n. HereCP (·) denotes complex projectivisation, ξk−1 is a complex line bundle over Bk−1

and C is a trivial line bundle. The fibre of the bundle pk : Bk → Bk−1 is CP 1.A Bott tower Bn is said to be topologically trivial if each pk : Bk → Bk−1 is

trivial as a smooth fibre bundle; in particular, such Bn is diffeomorphic to a productof 2-dimensional spheres.

We shall refer to the last stage Bn in a Bott tower as a Bott manifold (althoughit is also often called by the same name ‘Bott tower’).

In order to describe the cohomology ring of a Bott manifold, we need thefollowing general result:

Theorem 7.8.2 (see [297, Chapter V]). Let ξ be a complex n-dimensionalvector bundle over a finite cell complex X with complex projectivisation CP (ξ),and let u ∈ H2(CP (ξ)) be the first Chern class of the tautological line bundle overCP (ξ). The integral cohomology ring of CP (ξ) is the quotient of the polynomialring H∗(X)[u] on a generator u with coefficients in H∗(X) by the single relation

un − c1(ξ)un−1 + · · ·+ (−1)ncn(ξ) = 0.

Corollary 7.8.3. H∗(Bk) is a free module over H∗(Bk−1) on generators 1and uk, where uk is the first Chern class of the tautological line bundle over Bk =CP (C⊕ ξk−1). The ring structure is determined by the single relation

u2k = c1(ξk−1)uk.

For simplicity, we denote the element p∗k(uk−1) ∈ H2(Bk) by uk−1; similarly,we denote by ui each of the elements in H∗(Bk), k > i, which map to each other


by the homomorphisms p∗k. Each line bundle ξk−1 is determined by its first Chernclass, which can be written as a linear combination

c1(ξk−1) = a1ku1 + a2ku2 + · · ·+ ak−1,kuk−1 ∈ H2(Bk−1).

It follows that a Bott tower of height n is uniquely determined by the list of integersaij : 1 6 i < j 6 n, where

(7.38) u2k =

k−1∑

i=1

aikuiuk, 1 6 k 6 n.

The cohomology ring of Bn is the quotient of Z[u1, . . . , un] by relations (7.38).It is convenient to organise the integers aij into an upper triangular matrix,

(7.39) A =

−1 a12 · · · a1n0 −1 · · · a2n...

.... . .

...0 0 · · · −1

.

Example 7.8.4. Let n = 2. Then a Bott tower B2 → B1 is determined by aline bundle ξ1 over B1 = CP 1, i.e. B2 is a Hirzebruch surface (see Example 5.1.8).We have ξ1 = ηk for some k ∈ Z where ηk denotes the kth tensor power of thetautological line bundle over CP 1. The cohomology ring is given by

H∗(B2) = Z[u1, u2]/(u21, u

22 − ku1u2).

We have

CP (C⊕ ηk) ∼= CP (C⊕ ηk′ ) ⇔ k = k′ mod 2,

where ∼= denotes a diffeomorphism. This is proved by the following observation.First, note that CP (ξ) ∼= CP (ξ ⊗ η) for any complex vector bundle ξ and linebundle η. Let k′ − k = 2ℓ for some ℓ ∈ Z, then

CP (C⊕ ηk) ∼= CP ((C⊕ ηk)⊗ ηℓ) = CP (ηℓ ⊕ ηk+ℓ) ∼= CP (C⊕ ηk′ ),

where the last diffeomorphism is induced by the vector bundle isomorphism ηℓ ⊕ηk+ℓ ∼= C⊕ ηk′ , as both are plane bundles over CP 1 with equal Chern classes.

On the other hand, a cohomology ring isomorphism H∗(CP (C ⊕ ηk)) ∼=H∗(CP (C⊕ ηk′ )) implies that k = k′ mod 2 (an exercise).

This example shows that the cohomology ring determines the diffeomorphismtype of a Bott manifold Bn for n = 2. We may ask if this is true for arbitrary n;the questions of this sort are discussed in the last subsection.

Example 7.8.5. The bounded flag manifold BFn is a Bott manifold. By Propo-sition 7.7.8, BFn = CP (C⊕ρ01), where ρ01 is the line bundle over BFn−1 whose fiberover a bounded flag U is its first space U1. By Corollary 7.8.3, the ring structureof H∗(BFn) is determined by the relation u2n = c1(ρ

01)un. As it is clear from the

proof of Proposition 7.7.8, ρ01 is the tautological line bundle over BFn−1 (consid-ered as a complex projectivisation), so c1(ρ

01) = un−1. (Warning: the line bundle

ρ01 over BFn is not the pullback of the line bundle ρ01 over BFn−1 by the projectionpn : BFn → BFn−1, because p∗n(ρ

01) = ρ02.)


We therefore obtain the identity u2n = un−1un in H∗(BFn), and matrix (7.39)for the structure of the Bott tower on the bounded flag manifold has the form

A =

−1 1 0 · · · 00 −1 1 · · · 0...

.... . .

. . ....

0 0 0 · · · 10 0 0 · · · −1

.

As a corollary of this example we obtain the following characterisation ofbounded flag manifolds:

Proposition 7.8.6. The bounded flag manifold BFn is the Bott manifold whosetower structure is defined as follows: in each Bk = CP (C⊕ ξk−1) the bundle ξk−1

is the tautological line bundle over Bk−1 = CP (C⊕ ξk−2), for 2 6 k 6 n.

Bott towers as toric manifolds.

Theorem 7.8.7. The Bott manifold Bn corresponding to a matrix A givenby (7.39) is isomorphic to the toric manifold corresponding to the complete fan Σwith 2n one-dimensional cones generated by the vectors

a0k = ek, a1

k = −ek + ak,k+1ek+1 + · · ·+ aknen (1 6 k 6 n),

and 2n maximal cones generated by the sets of vectors aε11 , . . . ,aεnn , where εk = 0, 1.

Proof. Let Xn denote the toric manifold corresponding to the fan describedin the theorem. We may assume by induction that Xn−1 = Bn−1 (the base of theinduction is clear, as X1 = B1 = CP 1). By the construction of Section 5.4, themanifold Xn can be obtained as the quotient of

Un = (z1, . . . , z2n) ∈ C2n : |zk|2 + |zk+n|2 6= 0, 1 6 k 6 n ∼= (C2 \ 0)n

by the action of the group Gn ∼= (C×)n given by (5.3) (we have m = 2n here).Explicitly, the inclusion Gn → (C×)2n is given by

(t1, . . . , tn) 7→ (t1, t−a121 t2, . . . , t

−a1n1 t−a2n2 · · · t−an−1,n

n−1 tn, t1, t2, . . . , tn).

Observe that Un = Un−1 × (C2 \ 0), Gn = Gn−1 × C×, and the last factor C×

(corresponding to tn) acts trivially on Un−1. Therefore, we have

Xn = Un/Gn =(Un−1 × (C2 \ 0)/C×

)/Gn−1

= Un−1 ×Gn−1 CP1 = CP

(Un−1 ×Gn−1 (C⊕ C)

),

where Un−1 ×Gn−1 (C ⊕ C) is a complex 2-plane bundle over Un−1/Gn−1 = Xn−1

defined by the representation of the algebraic torus Gn−1∼= (C×)n−1 in C ⊕ C

which is given by the character (t1, . . . , tn−1) 7→ t−a1n1 t−a2n2 · · · t−an−1,n

n−1 on the firstsummand and is trivial on the second summand.

By the inductive assumption, Xn−1 = Bn−1. The bundle Un−1×Gn−1 (C⊕C) is

ξn−1⊕C, where ξn−1 is the line bundle overBn−1 with first Chern class∑n−1

i=1 ainui.Thus, Xn = CP (ξn−1 ⊕ C) = Bn, and the inductive step is complete.

Note that the associated simplicial complex of the fan described in Theo-rem 7.8.7 is the boundary of a cross-polytope, so the toric manifold is also a qua-sitoric manifold over a cube. We therefore obtain:


Corollary 7.8.8. A Bott tower of height n determined by matrix A has anatural action of the torus T n making it into a quasitoric manifold over a cubewith refined characteristic matrix Λ = (I | At), see (7.6).

Remark. The bounded flag manifold BFn has a toric structure described inProposition 7.7.3, and another toric structure coming from its Bott tower structurevia Theorem 7.8.7 (its matrix A is given in Example 7.8.5). The corresponding fansare not the same, but isomorphic (see Exercise 7.8.33).

Remark. The relations in the face ring of an n-cube are vivi+n = 0, 1 6 i 6 n.These relations together with (7.17) give the relations (7.38) after substitutionΛ = (I | At) and ui = −vi+n. In fact, one has ui = c1(ρi+n), where ρi+n isthe line bundle (7.7) over the toric manifold Bn (an exercise). It follows that thedescription of the cohomology ring of a Bott manifold from Corollary 7.8.3 agreeswith the description of the cohomology of a toric manifold from Theorem 7.3.27.

Given a permutation σ of n elements, denote by P (σ) the corresponding per-mutation matrix, the square matrix of size n with ones at the positions (i, σ(i))for 1 6 i 6 n, and zeros elsewhere. There is an action of the symmetric groupSn on square n-matrices by conjugations, A 7→ P (σ)−1AP (σ), or, equivalently, bypermutations of the rows and columns of A.

Proposition 7.8.9. A quasitoric manifold M over a cube with refined char-acteristic matrix Λ = (I | Λ⋆) is equivalent to a Bott manifold if and only if Λ⋆ isconjugate by means of a permutation matrix to an upper triangular matrix.

Proof. Assume that Λ⋆ is conjugate by means of a permutation matrix toan upper triangular matrix. Clearly, this condition is equivalent to the conjugacyof Λ⋆ to a lower triangular matrix. Consider the action of Sn on the set of facetsof the cube In by permuting pairs of opposite facets. A rearrangement of facetscorresponds to a rearrangement of columns in the characteristic n× 2n-matrix Λ,so an element σ ∈ Sn acts as

Λ 7→ Λ ·(P (σ) 00 P (σ)

).

This action does not preserve the refined form of Λ, as (I | Λ⋆) becomes(P (σ) | Λ⋆P (σ)). The refined representative in the left coset (7.5) of the lattermatrix is given by (I | P (σ)−1Λ⋆P (σ)). (In other words, we must compensate forthe permutation of pairs of facets by an automorphism of the torus Tn permutingthe coordinate subcircles to keep the characteristic matrix in the refined form.)This implies that the action by permutations on pairs of opposite facets induces anaction by conjugations on refined submatrices Λ⋆. Hence we may assume, up to anequivalence, that the refined characteristic submatrix Λ⋆ of M is lower triangular.The non-singularity condition (7.4) guarantees that the diagonal entries of Λ⋆ areequal to ±1, and we can set all of them equal to −1 by changing the omniorien-tation of M if necessary. Now, M has the same characteristic matrix as the Bottmanifold corresponding to the matrix A = Λt⋆ (see Corollary 7.8.8). Therefore, Mand the Bott manifold are equivalent by Proposition 7.3.11.

The converse statement follows from Corollary 7.8.8.

Our next goal is to characterise Bott manifolds within the class of quasitoricmanifolds over cubes more explicitly. Given a subset i1, . . . , ik ⊂ [n], the principal


minor of a square n-matrix A is the determinant of the submatrix formed bythe elements in columns and rows with numbers i1, . . . , ik. In the case of Bottmanifolds, according to Corollary 7.8.8, all principal minors of the matrix −Λ⋆ areequal to 1; for an arbitrary quasitoric manifold the non-singularity condition (7.4)only guarantees that all principal minors of Λ⋆ are equal to ±1.

Recall that an upper triangular matrix is unipotent if all its diagonal entriesare ones. The following key technical lemma can be retrieved from the proof ofDobrinskaya’s general result [96, Theorem 6] characterising quasitoric manifoldsover products of simplices which can be decomposed into towers of fibrations.

Lemma 7.8.10. Let R be a commutative integral domain with identity element 1,and let A be a square n-matrix (n > 2) with entries in R. Suppose that everyproper principal minor of A is equal to 1. If detA = 1, then A is conjugate bymeans of a permutation matrix to a unipotent upper triangular matrix, otherwiseit is permutation-conjugate to a matrix of the following form:

(7.40)

1 b1 0 . . . 00 1 b2 . . . 0...

.... . .

. . ....

0 0 . . . 1 bn−1

bn 0 . . . 0 1

where bi 6= 0 for all i.

Proof. By assumption the diagonal entries of A must be ones. We say thatthe ith row is elementary if its ith entry is 1 and the other entries are 0. Assumingby induction that the theorem holds for matrices of size (n−1) we deduce that A isitself permutation-conjugate to a unipotent upper triangular matrix if and only ifit contains an elementary row. We denote by Ai the square (n−1)-matrix obtainedby removing from A the ith column and the ith row.

We may assume by induction that An is a unipotent upper triangular matrix.Next we apply the induction assumption to A1. The permutation of rows andcolumns transforming A1 into a unipotent upper triangular matrix turns A into an‘almost’ unipotent upper triangular matrix; the latter may have only one non-zeroentry below the diagonal, which must be in the first column. If an1 = 0, then thenth row of A is elementary and A is permutation-conjugate to a unipotent uppertriangular matrix. Otherwise we have

A =

1 ∗ ∗ · · · ∗0 1 ∗ · · · ∗...

.... . .

. . ....

0 0 · · · 1 bn−1

bn 0 · · · 0 1

,

where bn−1 6= 0 and bn 6= 0 (otherwise A contains an elementary row). Now let a1j1be the last non-zero entry in the first row of A. If A does not contain an elementaryrow, then we may define by induction ajiji+1 as the last non-zero non-diagonal entryin the jith row of A. Clearly, we have

1 < j1 < · · · < ji < ji+1 < · · · < jk = n


for some k < n. Now, if ji = i + 1 for 1 6 i 6 n − 1, then A is the matrix (7.40)with bi = aji−1ji , 1 6 i 6 n− 1. Otherwise, the submatrix

S =

1 a1j1 0 . . . 00 1 aj1j2 . . . 0...

.... . .

. . ....

0 0 . . . 1 ajk−1n

bn 0 . . . 0 1

,

of A formed by the columns and rows with indices 1, j1, . . . , jk is proper and hasdeterminant 1± bn

∏ajiji+1 6= 1. This contradiction finishes the proof.

Theorem 7.8.11. Let M = M(In, Λ) be a quasitoric manifold over a cubewith canonical T n-invariant smooth structure, and Λ⋆ the corresponding refinedsubmatrix. Then the following conditions are equivalent:

(a) M is equivalent to a Bott manifold;(b) all the principal minors of −Λ⋆ are equal to 1;(c) M admits a T n-invariant almost complex structure (with the associated

omniorientation).

Proof. The implication (b) ⇒ (a) follows from Lemma 7.8.10 and Proposi-tion 7.8.9. The implication (a) ⇒ (c) is obvious. Let us prove (c) ⇒ (b). Recallthe definition of the sign σ(v) and the formula from Lemma 7.3.18 (b) expressingthis sign in terms of the combinatorial data. Denote the facets of the cube In byF ε1 , . . . , F

εn (ε = 0, 1), assuming that F 0

k ∩ F 1k = ∅, for 1 6 k 6 n. The normal

vectors of facets are aεk = (−1)εek. A vertex of In is given by

v = F ε11 ∩ · · · ∩ F εnn .

Therefore, the expression for the sign σ(v) on the right hand side of the formulafrom Lemma 7.3.18 is equal to a principal minor of the matrix −Λ⋆ (namely, theminor formed by the columns and rows with numbers i such that v ∈ F 1

i ). Itremains to note that in the almost complex case the sign of every vertex is 1.

Remark. The equivalence (a)⇔(b) is a particular case of [96, Theorem 6].

Corollary 7.8.12. Let V be a toric manifold whose associated fan is combina-torially equivalent to the fan consisting of cones over the faces of a cross-polytope.Then V is a Bott manifold.

Proof. If we view V as a quasitoric manifold (over a cube), then all theprincipal minors of the corresponding matrix Λ⋆ are equal to 1 by the same reasonas in the proof of Theorem 7.8.11. By Lemma 7.8.10, the matrix Λ⋆ is permutation-conjugate to a unipotent upper triangular matrix, so the full characteristic matrixΛ has the same form as the characteristic matrix of a Bott manifold. The columnsof Λ are the primitive vectors along edges of the fan corresponding to V , so thecombinatorial type of the fan and the matrix Λ determine the fan completely. Itfollows that the fan of V is the same as the fan of some Bott manifold, which impliesthat V has the structure of a Bott tower.

Example 7.8.13. Corollary 7.8.12 shows that the class of Bott manifolds co-incides with the class of toric manifolds over cubes, i.e. the first inclusion in thehierarchy described in the beginning of this section is an identity. This is not the


case for the second inclusion. For example the quasitoric manifold M over a square

with refined characteristic submatrix Λ⋆ =

(1 21 1

)is not a Bott manifold, be-

cause Λ⋆ is not permutation-conjugate to an upper triangular matrix. This M isdiffeomorphic to CP 2 #CP 2.

Semifree circle actions. Recall that an action of a group is called semifreeif it is free on the complement to fixed points. A particularly interesting class ofHamiltonian semifree circle actions was studied by Hattori, who proved in [152]that a compact symplectic manifold M carrying a semifree Hamiltonian S1-actionwith nonempty isolated fixed point set has the same cohomology ring and thesame Chern classes as CP 1 × · · · × CP 1, thus imposing a severe restriction onthe topological structure of the manifold. Hattori’s results were further extendedby Tolman and Weitsman, who showed in [309] that a semifree symplectic S1-action with nonempty isolated fixed point set is automatically Hamiltonian, andthe equivariant cohomology ring and Chern classes of M also agree with those ofCP 1 × · · · × CP 1. In dimensions up to 6 it is known that a symplectic manifoldwith a S1-action satisfying the properties above is diffeomorphic to a product of2-spheres, but in higher dimensions this remains open.

Ilinskii considered in [167] an algebraic version of Hattori’s question on semifreesymplectic S1-actions. Namely, he conjectured that a smooth compact complexalgebraic variety V carrying a semifree action of the algebraic 1-torus C× withpositive number of isolated fixed points is homeomorphic to S2 × · · · × S2. Thealgebraic and symplectic versions of the conjecture are related via the commonsubclass of projective varieties; a smooth projective variety is a symplectic manifold.Ilinskii proved the toric version of his algebraic conjecture, namely, when V is a toricmanifold and the semifree 1-torus is a subgroup of the acting torus (of dimensiondimC V ). The first step of Ilinskii’s argument was to show that if V admits asemifree action of a subcircle with isolated fixed points, then the correspondingfan is combinatorially equivalent to the fan over the faces of a cross-polytope. ByCorollary 7.8.12, such a toric manifold V admits a structure of a Bott tower.

The above described classification of Bott towers can therefore be applied toobtain results on semifree circle actions. According to a result of [210] (Theo-rem 7.8.16 below), a quasitoric manifold over a cube with a semifree circle actionis a Bott tower. By another result of [210], all such Bott towers are topologicallytrivial, i.e. diffeomorphic to a product of 2-dimensional spheres (Theorem 7.8.17).

A complex n-dimensional representation of the circle S1 is determined by the setof weights kj ∈ Z, 1 6 j 6 n. In appropriate coordinates an element s = e2πiϕ ∈ S1

acts as follows:

(7.41) s · (z1, . . . , zn) = (e2πik1ϕz1, . . . , e2πiknϕzn).

The following result is straightforward.

Proposition 7.8.14. A representation of S1 in Cn is semifree if and only ifkj = ±1 for 1 6 j 6 n.

Let M = M(P,Λ) be a quasitoric manifold. A circle subgroup in Tn is deter-mined by a primitive integer vector ν = (ν1, . . . , νn):

(7.42) S(ν) = (e2πiν1ϕ, . . . , e2πiνnϕ) ∈ Tn : ϕ ∈ R.


Given a vertex v = Fj1 ∩ · · · ∩ Fjn of P , we can decompose ν in terms of the basisλj1 , . . . , λjn :

(7.43) ν = k1(ν, v)λj1 + · · ·+ kn(ν, v)λjn .

Proposition 7.8.15. A circle S(ν) ⊂ Tn acts on a quasitoric manifold M =M(P,Λ) semifreely and with isolated fixed points if and only if, for every vertexv = Fj1 ∩ · · · ∩ Fjn , the coefficients in (7.43) satisfy ki(ν, v) = ±1 for 1 6 i 6 n.

Proof. It follows from Proposition 7.3.17 that the coefficients ki(ν, v) are theweights of the representation of the circle S(ν) in the tangent space to M at v. Thestatement follows from Proposition 7.8.14.

Theorem 7.8.16 ([210]). Let M be a quasitoric manifold over a cube In. As-sume that the torus acting on M has a circle subgroup acting semifreely and withisolated fixed points. Then M is equivalent to a Bott tower.

Proof. Let Λ⋆ be the refined characteristic submatrix of M . We may assumeby induction that every characteristic submanifold of M is a Bott manifold, so thatevery proper principal minor of the matrix −Λ⋆ is 1. Therefore, we are in thesituation of Lemma 7.8.10, and −Λ⋆ is a matrix of one of the two types describedthere. The second type is ruled out because of the semifreeness assumption. Indeed,let Λ = (I | −B), where B is the matrix (7.40) and assume that S(ν) ⊂ Tn actssemifreely with isolated fixed points. Applying the criterion from Proposition 7.8.15to the vertex v = F 0

1 ∩ · · · ∩ F 0n we obtain νi = ±1 for 1 6 i 6 n. Now we apply

the same criterion to the vertex v′ = F 11 ∩ · · · ∩F 1

n . Since the submatrix formed bythe corresponding columns of Λ is precisely −B, it follows that detB = ±1. Thisimplies that bi = ±1 in (7.40) for some i. Therefore, if all the coefficients kj(ν, v

′)in the expression ν = k1(ν, v

′)λn+1 + · · · + kn(ν, v′)λ2n are equal to ±1, then the

ith coordinate of ν is νi = ±1± bi 6= ±1: a contradiction.

Our next result shows that a Bott tower with a semifree circle subgroup andisolated fixed points is topologically trivial. Let t (respectively, C) be the standard(respectively, the trivial) complex one-dimensional representation of the circle S1,and let V denote the trivial bundle with fibre V over a given base. We say thatan action of a group G on a Bott manifold Bn preserves the tower structure if foreach stage Bk = CP (C⊕ ξk−1) the line bundle ξk−1 is G-equivariant. The intrinsicTn-action on Bn (see Corollary 7.8.8) obviously preserves the tower structure.

Theorem 7.8.17 ([210]). Assume that a Bott manifold Bn admits a semifreeS1-action with isolated fixed points preserving the tower structure. Then the Botttower is topologically trivial; furthermore, Bn is S1-equivariantly diffeomorphic tothe product (CP (C⊕ t))n.

Proof. We may assume by induction that the (n−1)th stage of the Bott toweris diffeomorphic to (CP (C ⊕ t))n−1 and Bn = CP (C ⊕ ξ) for some S1-equivariantline bundle ξ over (CP (C⊕ t))n−1.

Let γ be the canonical line bundle over CP (C⊕ t) ∼= CP 1. It carries a uniquestructure of an S1-equivariant bundle such that

(7.44) γ|(1:0) = C and γ|(0:1) = t.

We denote by x ∈ H2(CP (C⊕ t)) the first Chern class of γ, and let xi = π∗i (x) ∈

H2(CP (C ⊕ t)n−1) be the pullback of x by the projection πi onto the ith factor.


Then c1(ξ) =∑n−1

i=1 aixi for some ai ∈ Z. The S1-equivariant line bundles ξ and

⊗n−1i=1 π

∗i (γ

ai) have the same underlying bundles, so there is an integer k such that

(7.45) ξ = tk ⊗n−1⊗

i=1

π∗i (γ

ai)

as S1-equivariant line bundles (see [155, Corollary 4.2]).We encode S1-fixed points in CP (C⊕t)n−1 by sequences (pε11 , . . . , p

εn−1

n−1 ), whereεi = 0 or 1, and pεii denotes (1 : 0) if εi = 0 and (0 : 1) if εi = 1. Then it followsfrom (7.44) and (7.45) that

ξ|(p

ε11 ,...,p

εn−1n−1 )

= tk+∑n−1

i=1 εiai .

The S1-action on Bn = CP (C ⊕ ξ) is semifree if and only if |k +∑n−1

i=1 εiai| = 1for all possible values of εi. Setting εi = 0 for all i we obtain |k| = 1. Letk = 1 (the case k = −1 is treated similarly). Then (a1, . . . , an−1) = (0, . . . , 0) or(0, . . . , 0,−2, 0, . . . , 0). In the former case, ξ = t and Bn = CP (C⊕ξ) ∼= CP (C⊕t)n.In the latter case we have ξ = tπ∗

i (γ−2) for some i, so that Bn = π∗

i CP (C⊕ tγ−2).Since for any S1-vector bundle E and S1-line bundle η the projectivisations CP (E)and CP (E⊗η) are S1-diffeomorphic, it follows that CP (C⊕ tγ−2) ∼= CP (γ⊕ tγ−1).The first Chern class of γ ⊕ tγ−1 is zero, so its underlying bundle is trivial. TheS1-representation in the fibre of γ ⊕ tγ−1 over a fixed point is isomorphic to C⊕ tby (7.44). Theorefore, γ ⊕ tγ−1 = C ⊕ t as S1-bundles. It follows that CP (C ⊕tγ−2) ∼= CP (C⊕ t) and Bn ∼= (CP (C⊕ t))n.

Remark. The diffeomorphism of Theorem 7.8.17 is not Tn-equivariant.

The next example shows that Theorem 7.8.17 cannot be generalised to qua-sitoric manifolds. However, as we shall see, it holds under the additional assumptionthat the quotient polytope of the quasitoric manifold is a cube.

Example 7.8.18. Let M be a quasitoric manifold over a 2k-gon with

Λ =

(1 0 1 0 · · · 1 00 1 0 1 · · · 0 1

).

By Corollary 7.8.15, the circle subgroup determined by the vector ν = (1, 1) actssemifreely on M . However, the quotient of M is not a 2-cube if k > 2, so M cannotbe homeomorphic to a product of spheres (it can be shown that M is a connectedsum of k − 1 copies of S2 × S2).

Theorem 7.8.19 ([210]). Let M a quasitoric manifold M over a cube In.Assume that the torus acting on M has a circle subgroup acting semifreely. ThenM is S1-equivariantly homeomorphic to a product of 2-spheres.

Proof. By Theorem 7.8.16, M is equivalent to a Bott tower. By Theo-rem 7.8.17, it is S1-homeomorphic to a product of spheres.

We can also derive Ilinskii’s result on semifree actions on toric manifolds:

Theorem 7.8.20 ([167]). A toric manifold V carrying a semifree action of acircle subgroup with isolated fixed points is diffeomorphic to a product of 2-spheres.


Proof. By Theorem 7.8.17, it is sufficient to show that V is a Bott manifold.A semifree circle subgroup acting on V also acts semifreely and with isolated fixedpoints on every characteristic submanifold Vj of V . We use induction on the di-mension. The base of induction is the case dimC V = 2; we consider it below. Bythe inductive hypothesis, each Vj is a Bott manifold, i.e. its quotient polytope isa combinatorial cube. On the other hand, the quotient polytope of Vj is the facetFj of the quotient polytope P of V ; since each Fj is a cube, P is also a cube byExercise 1.1.22. By Corollary 7.8.12, V is a Bott manifold, and we are done.

It remains to consider the case dimC V = 2. We need to show that the quotientpolytope of a complex 2-dimensional toric manifold V with semifree circle subgroupaction and isolated fixed points is a 4-gon.

Let Σ be the fan corresponding to V . One-dimensional cones of Σ correspond tofacets (or edges) of the quotient polygon P 2. We must show that there are precisely4 one-dimensional cones. The values of the characteristic function on the facets ofP 2 are given by the primitive vectors generating the corresponding one-dimensionalcones of Σ. Let ν be the vector generating the semifree circle subgroup. We maychoose an initial vertex v of P 2 so that ν belongs to the 2-dimensional cone of Σcorresponding to v. Then we index the primitive generators a i, 1 6 i 6 m, of1-cones so that ν is in the cone generated by a1 and a2, and any two consecutivevectors span a two-dimensional cone (see Fig. 7.6). This provides us with a refined

a1

ννa2

a3

am

· ··

Figure 7.6.

characteristic matrix Λ of size 2 × m. We have a1 = (1, 0) and a2 = (0, 1), andapplying the criterion from Proposition 7.8.15 to the first cone R>〈a1,a2〉 (that is,to the initial vertex of the polygon) we obtain ν = (1, 1).

The rest of the proof is a case by case analysis using the non-singularity con-dition (7.4) and Proposition 7.8.15. The reader may be willing to do this himselfas an exercise rather than following the argument below.

Consider now the second cone. The non-singularity conditions (7.4) gives usdet(a2,a3) = 1, hence a3 = (−1, ∗). Writing νν = k1a2 + k2a3 and applyingProposition 7.8.15 to the second cone R>〈a2,a3〉 we obtain

(1, 1) = ±(0, 1)± (−1, ∗).Therefore, a3 = (−1, 0) or a3 = (−1,−2). Similarly, considering the last coneR>〈am,a1〉 we obtain am = (∗,−1), and then, applying Proposition 7.8.15, wesee that am = (0,−1) or am = (−2,−1). The case when a3 = (−1,−2) andam = (−2,−1) is impossible since then the second and the last cones overlap.


Let a3 = (−1,−2). Then am = (0,−1). Considering the cone R>〈am−1,am〉we obtain am−1 = (−1, 0) or am−1 = (−1,−2). In the former case cones overlap,and in the latter case we get am−1 = a3. This implies m = 4 and we are done.

Let a3 = (−1, 0). Then considering the third cone R>〈a3,a4〉 we obtain a4 =(0,−1) or a4 = (−2,−1). If a4 = (0,−1), then a4 = am (otherwise cones overlap),and we are done. Let a4 = (−2,−1). Then either am = (−2,−1) = a4 and weare done, or am = (0,−1). In the latter case, we get am−1 = (−1,−2) (see theprevious paragraph).

We are left with the case a4 = (−2,−1) and am−1 = (−1,−2). The only wayto satisfy both (7.4) and the condition of Proposition 7.8.15 without overlappingcones is to set a5 = (−3,−2) and am−2 = (−2,−3). Continuing this process, weobtain ak = (−k+ 2,−k+ 3), k > 2, and am−l = (−l,−l− 1), l > 0. This processnever stops, as we never get a complete regular fan. So this case is impossible.

The proof above leaves three possibilities for the vectors a3 and a4 of the2-dimensional fan: (−1, 0) and (0,−1), or (−1, 0) and (−2,−1), or (−1,−2) and(0,−1). The last two pairs correspond to isomorphic fans. The refined characteristicsubmatrices corresponding to the first two pairs are(

−1 00 −1

)and

(−1 −20 −1

).

The first corresponds to CP 1×CP 1, and the second to a Bott manifold (Hirzebruchsurface) with a12 = −2.

The next result gives an explicit description of matrices (7.39) correspondingto our specific class of Bott towers:

Theorem 7.8.21 ([210]). A Bott manifold Bn admits a semifree circle subgroupwith isolated fixed points if and only if its matrix (7.39) satisfies the identity

1

2(E −A) = C1C2 · · ·Cn,

where Ck is either the identity matrix or a unipotent upper triangular matrix withonly one nonzero element above the diagonal; this element is 1 in the kth column.

Proof. Assume first that Bn admits a semifree circle subgroup with isolatedfixed points. We have two sets of multiplicative generators for the ring H∗(Bn):the set u1, . . . , un from Corollary 7.8.3 satisfying identities (7.38), and the setx1, . . . , xn satisfying x2i = 0 (the latter set exists since Bn is diffeomorphicto a product of 2-spheres). The reduced sets with i 6 k can be regarded asthe corresponding sets of generators for the kth stage Bk. As is clear from theproof of Theorem 7.8.17, we have c1(ξk−1) = −2cikkxik for some ik < k, wherecikk = 1 or 0. From u2k + 2cikkxikuk = 0 we obtain xk = uk + cikkxik . Inother words, the transition matrix Ck from the basis x1, . . . , xk−1, uk, . . . , un ofH2(Bn) to x1, . . . , xk, uk+1, . . . , un may have only one nonzero entry off the di-agonal, which is cikk. The transition matrix from u1, . . . , un to x1, . . . , xn is theproduct D = C1C2 · · ·Cn. Then D = (djk) is a unipotent upper triangular matrixconsisting of zeros and ones, xk =

∑nj=1 djkuj and

0 = x2k =(uk +

∑k−1j=1 djkuj

)2= u2k + 2

∑k−1j=1 djkujuk + · · · , 1 6 k 6 n.

On the other hand, 0 = u2k−∑k−1j=1 ajkujuk by (7.38). Comparing the coefficients of

ujuk for 1 6 j 6 k− 1 in the last two equations and observing that these elements


are linearly independent in H4(B2k) we obtain 2djk = −ajk for 1 6 j < k 6 n. Asboth D and −A are unipotent upper triangular matrices, this implies 2D = E−A.

Assume now that the matrix A satisfies E − A = 2C1C2 · · ·Cn. Then for thecorresponding Bott tower we have ξk−1 = π∗

ik(γ−2cikk). Therefore, we may choose a

circle subgroup such that ξk−1 becomes tπ∗ik(γ−2cikk) (as an S1-equivariant bundle),

for 1 < k 6 n. This circle subgroup acts semifreely and with isolated fixed pointsas seen from the same argument as in the proof of Theorem 7.8.17.

Example 7.8.22. The condition of Theorem 7.8.21 implies in particular thatthe matrix (7.39) may have only entries equal to 0 or −2 above the diagonal.However, the hypothesis of Theorem 7.8.21 is stronger. For instance, if

A =

−1 0 −20 −1 −20 0 −1

then the matrix (E − A)/2 cannot be factored as C1C2C3. Consequently, thecorresponding 3-stage Bott tower does not admit a subcircle acting semifreely andwith isolated fixed points. On the other hand, if

A =

−1 −2 −20 −1 00 0 −1

then we have

1

2(E −A) =

1 1 10 1 00 0 1

=

1 0 00 1 00 0 1

1 1 00 1 00 0 1

1 0 10 1 00 0 1

.

It is clear that not every topologically trivial Bott manifold admits a semifreesubcircle action with isolated fixed points (the latter condition is stronger even forn = 2). We now consider the former class in more detail.

Topologically trivial Bott towers. The topological triviality of a Bott towercan be detected by its cohomology ring:

Theorem 7.8.23 ([210]). A Bott tower Bn is topologically trivial if and onlyif there is an isomorphism H∗(Bn) ∼= H∗((S2)n) of graded rings.

Proof. Corollary 7.8.3 implies

H∗(Bn) = H∗(Bn−1

)[un]

/(u2n − c1(ξn−1)un

).

We may therefore write any element of H2(Bn) as x + bun, where x ∈ H2(Bn−1)and b ∈ Z. We have

(x+ bun)2 = x2 + 2bxun + b2u2n = x2 + b(2x+ bc1(ξn−1))un,

so that the square of x + bun with b 6= 0 is zero if and only if x2 = 0 and 2x +bc1(ξn−1) = 0. This shows that elements of the form x + bun with b 6= 0 whosesquares are zero generate a rank-one free subgroup of H2(Bn).

Assume that H∗(Bn) ∼= H∗((CP 1)n). Then there is a basis x1, . . . , xn inH2(Bn) such that x2i = 0 for all i. By the observation from the previous paragraph,we may assume that the elements x1, . . . , xn−1 lie in H2(Bn−1), and xn is not in

H2(Bn−1). Then we can have xn =∑n−1

i=1 bixi + un for some bi ∈ Z. A product∏i∈I xi with I ⊂ 1, . . . , n lies in H∗(Bn−1) if and only if n /∈ I. This implies that


the ring H∗(Bn−1) is generated by the elements x1, . . . , xn−1 and is isomorphic tothe cohomology ring of (CP 1)n−1. Therefore, we may assume by induction thatBn−1

∼= (CP 1)n−1.

Writing c1(ξn−1) =∑n−1i=1 aixi, we obtain

0 = x2n =(un +

∑n−1i=1 bixi

)2=∑n−1

i=1 (ai + 2bi)xiun +(∑n−1

i=1 bixi)2.

This may hold only if at most one of the ai is nonzero (and equal to −2bi) becausethe elements xixj and xiun with i < j < n form a basis of H4(Bn). Therefore, ξn−1

is the pullback of the bundle γ−2bi over CP 1 by the ith projection map Bn−1 =(CP 1)n−1 → CP 1. Since CP (C ⊕ γ−2bi) is a topologically trivial bundle (seeExample 7.8.4), the bundle Bn = CP (C⊕ ξn−1) is also trivial.

We can now also describe effectively the class of matrices (7.39) correspondingto topologically trivial Bott towers:

Theorem 7.8.24. A Bott tower is topologically trivial if and only if its corre-sponding matrix (7.39) satisfies the identity

1

2(E −A) = C1C2 · · ·Cn,

where Ck is either the identity matrix or a unipotent upper triangular matrix withonly one nonzero element above the diagonal; this element lies in the kth column.

Proof. The argument is the same as in the proof of Theorem 7.8.21. The onlydifference is that the number cikk in the formula c1(ξk−1) = −2cikkxik is now anarbitrary integer.

Theorem 7.8.23 can be generalised to quasitoric manifolds, but only in thetopological category:

Theorem 7.8.25 ([210, Theorem 5.7]). A quasitoric manifold M is homeomor-phic to a product (S2)n if and only if there is an isomorphism H∗(M) ∼= H∗((S2)n)of graded rings.

Generalisations and cohomological rigidity.

Definition 7.8.26. A generalised Bott tower of height n is a tower of bundles

Bnpn−→ Bn−1

pn−1−→ · · · −→ B2p2−→ B1 −→ pt ,

of complex manifolds, whereB1 = CP j1 and eachBk is the complex projectivisationof a sum of jk+1 complex line bundles over Bk−1. The fibre of the bundle pk : Bk →Bk−1 is CP jk . A generalised Bott tower is topologically trivial if each pk is trivialas a smooth bundle.

The last stage Bn in a generalised Bott tower is a generalised Bott manifold.

Remark. A generalised Bott manifold Bn is a projective toric manifold whosecorresponding polytope is combinatorially equivalent to a product of simplices∆j1×· · · ×∆jn (an exercise). In particular, Bn is a quasitoric manifold over a productof simplices. If we replace ‘a sum of complex line bundles’ by ‘a complex vectorbundle’ in the definition, then the resulting tower will not be a toric manifold ingeneral: the torus action on Bk−1 lifts to the projectivisation of a sum of linebundles, but not to the projectivisation of an arbitrary vector bundle over Bk−1.


Generalised Bott towers were considered by Dobrinskaya [96], who proved thefollowing result (see Lemma 7.3.18 for the information about the signs of vertices):

Theorem 7.8.27 ([96, Corollary 7]). A quasitoric manifold over a product ofsimplices P = ∆j1 × · · · ×∆jn is a generalised Bott manifold if and only if the signof each vertex of P is the product of the signs of its corresponding vertices of thesimplices ∆jk according to the decomposition of P . In particular, a toric manifoldover a product of simplices is always a generalised Bott manifold.

Remark. According to the general result [96, Theorem 6], a quasitoric man-ifold over any product polytope P = Pj1 × · · · × Pjn decomposes into a tower ofquasitoric fibre bundles if and only if the sign of each vertex of P decomposes intothe corresponding product.

Theorem 7.8.23 can be extended to generalised Bott towers:

Theorem 7.8.28 ([75, Theorem 1.1]). If the integral cohomology ring of ageneralised Bott manifold Bn is isomorphic to H∗(CP j1 × · · · × CP jn), then thegeneralised Bott tower is topologically trivial; in particular, Bn is diffeomorphic toCP j1 × · · · × CP jn .

According to another result of Choi, Masuda and Suh [74, Theorem 8.1], a qua-sitoric manifold M over a product of simplices P = ∆j1×· · ·×∆jn is homeomorphicto a generalised Bott manifold if H∗(M) ∼= H∗(CP j1×· · ·×CP jn). Therefore, sucha quasitoric manifold M is homeomorphic to CP j1 × · · · × CP jn .

Since trivial (generalised) Bott towers can be detected by their cohomologyrings, a natural question arises of whether any two (generalised) Bott manifoldscan be distinguished, either smoothly or topologically, by their cohomology rings.This leads to the notion of cohomological rigidity:

Definition 7.8.29. Fix a commutative ring k with unit. We say that a fam-ily of closed manifolds is cohomologically rigid over k if manifolds in the familyare distinguished up to homeomorphism by their cohomology rings with coeffi-cients in k. That is, a family is cohomologically rigid if a graded ring isomorphismH∗(M1;k) ∼= H∗(M2;k) implies a homeomorphism M1

∼= M2 whenever M1 andM2 are in the family.

A manifold M in the given family is said to be cohomologically rigid if for anyother manifold M ′ in the family a ring isomorphism H∗(M ;k) ∼= H∗(M ′;k) impliesa homeomorphism M ∼=M ′. Obviously a family is cohomologically rigid wheneverevery its element is rigid.

There is a smooth version of cohomological rigidity for families of smooth man-ifolds, with homeomorphisms replaced by diffeomorphisms.

Problem 7.8.30. Is the family of Bott manifolds cohomologically rigid(over Z)? Namely, is it true that any two Bott manifolds M1 and M2 with isomor-phic integral cohomology rings are homeomorphic (or even diffeomorphic)?

This question is open even for the much larger families of toric and quasitoricmanifolds. Namely, there are no known examples of non-homeomorphic (quasi)toricmanifolds with isomorphic integral cohomology rings. By the result of Choi, Parkand Suh [77], quasitoric manifolds with second Betti number 2 (i.e. over a productof two simplices) are homeomorphic when their cohomology rings are isomorphic.


In the positive direction, Theorem 7.8.23 shows that a topologically trivial Bottmanifold is cohomologically rigid in the family of Bott manifolds, in the smooth cat-egory. By Theorem 7.8.25, a topologically trivial Bott manifold is cohomologicallyrigid in the wider family of quasitoric manifolds, but only in the topological cate-gory. Smooth cohomological rigidity was established in Choi [71] for Bott manifoldsup to dimension 8 (i.e. up to height 4).

There is an R-version of this circle of questions, with quasitoric manifolds re-placed by small covers , (generalised) Bott towers replaced by real (generalised) Botttowers, and the cohomology rings taken with coefficients in Z2. A real generalisedBott tower is defined similarly to a complex tower, with the complex projectivisa-tion replaced by the real one.

The family of real Bott manifolds is cohomologically rigid over Z2 by the resultof Kamishima–Masuda [178] (see also [73]), but the family of generalised Bottmanifolds is not [208]. Also, every small cover over a product of simplices isa generalised real Bott tower [74] (this is not true for quasitoric manifolds, seeExample 7.8.13). For more results on the cohomological rigidity of (generalised)Bott towers, see the survey articles [211] and [76].

Exercises.

7.8.31. Let ηk denote the kth tensor power of the tautological line bundleover CP 1, and C denote the trivial line bundle. Show that there is a cohomologyring isomorphismH∗(CP (C⊕ηk)) ∼= H∗(CP (C⊕ηk′)) if and only if k = k′ mod 2.

7.8.32. Show that the toric manifold described in Theorem 7.8.7 is projectiveby providing explicitly a polytope P whose normal fan is Σ. Observe that P iscombinatorially equivalent to a cube (compare Proposition 7.7.4).

7.8.33. The bounded flag manifold BFn is a Bott tower, as described in Ex-ample 7.8.5. By Theorem 7.8.7, it is isomorphic to the toric manifold whose corre-sponding fan has generators

a0k = ek (1 6 k 6 n), a1

k = −ek + ek+1 (1 6 k 6 n− 1), a1n = en.

This fan is not the same as the one described in Proposition 7.7.3. However, thereis a linear isomorphism of Rn taking one fan to another, and therefore the corre-sponding toric manifolds are isomorphic.

7.8.34. Let Bn be the Bott manifold and ui ∈ H2(Bn) the canonical cohomol-ogy ring generator (obtained by pulling back the first Chern class of the tautologicalline bundle over Bi to the top stage Bn). Show that ui = c1(ρi+n), where ρi+n isthe line bundle (7.7) over the toric manifold Bn. (Hint: use induction.)

7.8.35. A generalised Bott manifold is a projective toric manifold over a productof simplices.

7.8.36. The projectivisation of a complex k-plane bundle (k > 1) over CPn isnot necessarily a toric manifold.

7.9. Weight graphs

The quotient space M2n/T n of a half-dimensional torus action has the orbitstratification (the face structure), therefore providing a natural combinatorial ob-ject associated with the action and allowing us to translate equivariant topology

7.9. WEIGHT GRAPHS 309

into combinatorics. For the classes of half-dimensional torus actions consideredin the previous sections, the quotient M2n/T n is contractible or acyclic, so manytopological invariants of the action can be expressed in purely combinatorial terms.

In this section we consider another type of combinatorial objects associated toT -manifolds, which catch some important information about the T -action and itsorbit structure: the so-called weight graphs (for particular classes of T -manifoldsthese are also known as GKM-graphs). A weight graph is an oriented graph with aspecial labelling of edges, which can be assigned to an effective T k-action on M2n

(k 6 n) with isolated fixed points under very mild assumption on the action. Whenk < n the topology of the quotient M2n/T k is often quite complicated, and theweight graph can be viewed as its combinatorial approximation.

In the study of quasitoric manifolds we constructed the correspondence (7.14),which assigns to each oriented edge of the quotient simple polytope P a weightof the T -representation at the fixed point corresponding to the origin of the edge.This correspondence can be viewed as a graph Γ with special labels on its ori-ented edges, and we refer to such an object as a weight graph. As it follows fromProposition 7.3.17, defining the correspondence (7.14) is equivalent to defining thecharacteristic matrix Λ. At the same time, it is well-known that the 1-skeletonof a simple polytope determines its entire combinatorial structure (see e.g. [325,Theorem 3.12]). It follows that the weight graph contains the same information asthe combinatorial quasitoric pair (P,Λ), and therefore it completely determines thetorus action on the quasitoric manifold M . For more general classes of torus actionsconsidered in this chapter, one cannot expect that the weight graph determines theaction, but it still contains an important piece of information.

Graphs whose oriented edges are labelled by the weights of a torus actionwere considered in the works of Musin [236], Hattori and other authors since the1970s. A renewed interest to these graphs was stimulated by the works of Goresky–Kottwitz–MacPherson [129] and Guillemin–Zara [145] in connection with the studyof symplectic manifolds with Hamiltonian torus actions. A related more generalclass of T -manifolds has become known as GKM-manifolds , and their weight graphsare often referred to as GKM-graphs.

A closed 2n-dimensional manifold M with an effective smooth action of torusT k (k 6 n) is called a GKM-manifold if the fixed point set is finite and nonempty,a T k-invariant almost complex structure is given on M , and the weights of thetangential T k-representation at any fixed point are pairwise linearly independent.As in the case of quasitoric manifolds, there is a weight graph associated with eachGKM-manifold M (vertices of the graph correspond to fixed points of M , and edgescorrespond to connected components of the set of points ofM with codimension-onestabilisers). As it was shown in [129], many important topological characteristicsof a GKM-manifold M (such as the Betti numbers or the equivariant cohomology)can be described in terms of the weight graph. Axiomatisation of the properties ofthe weight graph of a GKM-manifold led to the notion of a GKM-graph [145].

Weight graphs arising from locally standard T -manifolds (or torus manifolds)were studied in [204]. As in the case of GKM-manifolds, axiomatisation of theproperties of weight graphs leads to an interesting combinatorial object, known asa T -graph (or torus graph).

A T -graph is a finite n-valent graph Γ (without loops, but with multiple edgesallowed) with an axial function on the set E(Γ) of oriented edges taking values in


Hom(T n, S1) = H2(BT n) and satisfying certain compatibility conditions. Theseconditions (described below) are similar to those for GKM-graphs, but not exactlythe same. The weight graph of a torus manifold is an example of a T -graph; in thiscase the values of the axial function are the weights of the tangential representationsof T n at fixed points.

The equivariant cohomology ring H∗T (Γ) of a T -graph Γ can be defined in

the same way as for GKM-graphs; when the T -graph arises from a locally standardtorus manifoldM we haveH∗

T (Γ) = H∗T (M). Furthermore, unlike the case of GKM-

graphs, the equivariant cohomology ring of a T -graph can be described in termsof generators and relations (see Theorem 7.9.12). Such a description is obtainedby defining the simplicial poset S(Γ) associated with a T -graph Γ; then H∗

T (Γ) isshown to be isomorphic to the face ring Z[S(Γ)]. This theorem continues the seriesof results identifying the equivariant cohomology of a (quasi)toric manifold [90] anda locally standard T -manifold (Theorem 7.4.33) with the face ring of the associatedpolytope, simplicial complex, or simplicial poset.

Although the classes of GKM- and T -graphs diverge in general, they containan important subclass of n-independent GKM-graphs in their intersection.

Definition of a T -graph. This definition is a natural adaptation of the notionof GKM-graph [145] to torus manifolds.

Let Γ be a connected n-valent graph without loops but possibly with multipleedges. Denote by V (Γ) the set of vertices and by E(Γ) the set of oriented edges(so that each edge enters E(Γ) twice with the opposite orientations). We furtherdenote by i(e) and t(e) the initial and terminal points of e ∈ E(Γ), respectively,and denote by e the edge e with the reversed orientation. For v ∈ V (Γ) we set

E(Γ)v = e ∈ E(Γ): i(e) = v.A collection θ = θe of bijections

θe : E(Γ)i(e) → E(Γ)t(e), e ∈ E(Γ),

is called a connection on Γ if

(a) θe is the inverse of θe;(b) θe(e) = e.

An n-valent graph Γ with g edges admits ((n− 1)!)g different connections. LetT = T n be an n-torus. A map

α : E(Γ)→ Hom(T, S1) = H2(BT )

is called an axial function (associated with the connection θ) if it satisfies thefollowing three conditions:

(a) α(e) = ±α(e);(b) elements of α(E(Γ)v) are pairwise linearly independent (2-independent)

for each vertex v ∈ V (Γ);(c) α(θe(e

′)) ≡ α(e′) mod α(e) for any e ∈ E(Γ) and e′ ∈ E(Γ)i(e).

We also denote by Te = kerα(e) the codimension-one subtorus in T determinedby α and e. Then we may reformulate the condition (c) above as follows: therestrictions of α(θe(e

′)) and α(e′) to H∗(BTe) coincide.

Remark. Guillemin and Zara required α(e) = −α(e) in their definition ofaxial function. A connection θ satisfying condition (c) above is unique if elementsof α(E(Γ)v) are 3-independent for each vertex v (an exercise, see [145]).


Definition 7.9.1. We call α a T -axial function if it is n-independent, i.e. ifα(E(Γ)v) is a basis of H2(BT ) for each v ∈ V (Γ). A triple (Γ, θ, α) consisting ofa graph Γ, a connection θ and a T -axial function α is called a T -graph. Since aconnection θ is uniquely determined by α, we often suppress it in the notation.

Remark. Compared with GKM-graphs, the definition of a T -graph has weakercondition (a) (we only require α(e) = ±α(e) instead of α(e) = −α(e)), but strongercondition (b) (α is required to be n-independent rather than 2-independent).

Example 7.9.2. Let M be a locally standard torus manifold. Denote by ΓMthe 1-skeleton of the orbit space Q =M/T (it is easy to see that the 1-skeleton canbe defined without the local standardness assumption), and let αM be the axialfunction of Lemma 7.4.24. Then (ΓM , αM ) is a T -graph.

Example 7.9.3. Two T -graphs are shown in Fig. 7.7. The first is 2-valent andthe second is 3-valent. The axial function α takes the edges, regardless of theirorientation, to the generators t1, t2 ∈ H2(BT 2) (respectively, t1, t2, t3 ∈ H2(BT 3)).These T -graphs are not GKM-graphs, as the condition α(e) = −α(e) is not satisfied.Both come from torus manifolds, S4 and S6, respectively (see Example 7.4.16).

r

r

r

rt1 t2 t1 t2 t3

(a) n = 2 (b) n = 3

Figure 7.7. T -graphs.

Definition 7.9.4. The equivariant cohomology H∗T (Γ) of a T -graph Γ is a set

of mapsf : V (Γ)→ H∗(BT )

such that for every e ∈ E(Γ) the restrictions of f(i(e)) and f(t(e)) to H∗(BTe)coincide. Since H∗(BT ) is a ring, the set of maps from V (Γ) to H∗(BT ), denotedby H∗(BT )V (Γ), is also a ring with respect to the vertex-wise multiplication. Itssubspace H∗

T (Γ) is a subring because the restriction map H∗(BT ) → H∗(BTe) ismultiplicative. Furthermore, H∗

T (Γ) is a H∗(BT )-algebra.

If M is a torus manifold withHodd(M) = 0, then for the corresponding T -graphΓM we have H∗

T (ΓM ) ∼= H∗T (M) by Theorem 7.4.28.

Calculation of equivariant cohomology. Here we interpret the results ofSection 7.4 on equivariant cohomology of torus manifolds in terms of their associ-ated T -graphs, thereby providing a purely combinatorial model for this calculation,which is applicable to a wider class of objects.

Definition 7.9.5. Let (Γ, θ, α) be a T -graph and Γ′ a connected k-valent sub-graph of Γ, where 0 6 k 6 n. If Γ′ is invariant under the connection θ, then we saythat (Γ′, α|E(Γ′)) is a k-dimensional face of Γ. As usual, (n− 1)-dimensional facesas called facets.


An intersection of faces is invariant under the connection, but can be discon-nected. In other words, such an intersection is a union of faces.

The Thom class of a k-dimensional face G = (Γ′, α|E(G′)) is the mapτG: V (Γ)→ H2(n−k)(BT ) defined by

(7.46) τG(v) =

∏

i(e)=v, e/∈Γ′

α(e) if v ∈ V (Γ′),

0 otherwise.

Lemma 7.9.6. The Thom class τG

is an element of H∗T (Γ).

Proof. Let e ∈ E(Γ). If neither of the vertices of e is contained in G, thenthe values of τ

Gon both vertices of e are zero. If only one vertex of e, say i(e),

is contained in G, then τG(t(e)) = 0, while τ

G(i(e)) = 0 mod α(e), so that the

restriction of τG(i(e)) to H∗(BTe) is also zero. Finally, assume that the whole

e is contained in G. Let e′ be an edge such that i(e′) = i(e) and e′ /∈ G, sothat α(e′) is a factor in τ

G(i(e)). Since G is invariant under the connection, it

follows that θe(e′) /∈ G. Therefore, α(θe(e

′)) is a factor in τG(t(e)). Now we have

α(θe(e′)) ≡ α(e′) mod α(e) by the definition of axial function. The same holds

for any other factor in τG(i(e)), whence the restrictions of τ

G(i(e)) and τ

G(t(e)) to

H∗(BTe) coincide.

Lemma 7.9.7. If Γ is a T -graph, then there is a unique k-face containing anygiven k elements of E(Γ)v.

Proof. Let S ⊂ E(Γ)v be the set of given oriented k edges with the commonorigin v. Consider the graph Γ′ obtained by ‘spreading’ S using the connection θ.In more detail, at the first step we add to S all oriented edges of the form θe(e

′)where e, e′ ∈ S. Denote the resulting set by S1. At the second step we add to S1

all oriented edges of the form θe(e′) where e, e′ ∈ S1 and i(e) = i(e′), an so on.

Since Γ is a finite graph, this process stabilises after a finite number of steps, andwe obtain a subgraph Γ′ of Γ. This subgraph Γ′ is obviously θ-invariant. We claimthat Γ′ is k-valent. To see this, define for any vertex w ∈ V (Γ′) the subgroup

Nw = Z〈α(e) : e ∈ E(Γ′)w〉 ⊂ H2(BT ).

Condition (c) from the definition of the axial function implies that Nw = Nw′ forany vertices w,w′ ∈ V (Γ′). Since Nv ∼= Zk for the initial vertex v, it follows thatNw ∼= Zk for any w ∈ V (Γ′). Now, then n-independence of the axial functionimplies that there are exactly k edges in the set e ∈ E(Γ′)w, for any vertex wof Γ′. In other words, Γ′ is k-valent and therefore it defines a k-face of Γ.

Corollary 7.9.8. Faces of a T -graph Γ form a simplicial poset S(Γ) of rankn with respect to reversed inclusion.

Denote by G ∨ H a minimal face containing both G and H . In general sucha least upper bound may fail to exist or be non-unique; however it exists and isunique provided that the intersection G ∩H is non-empty.

Lemma 7.9.9. For any two faces G and H of Γ the corresponding Thom classessatisfy the relation

(7.47) τGτH= τ

G∨H·∑

E∈G∩H

τE,


where we formally set τΓ= 1 and τ∅ = 0, and the sum in the right hand side is

taken over connected components E of G ∩H.

Proof. We need to check that both sides of the identity take the same valueson any vertex v. The argument is the same as in the proof of Lemma 7.4.26.

Lemma 7.9.10. The Thom classes τG

corresponding to all proper faces of Γconstitute a set of ring generators for H∗

T (Γ).

Proof. This is proved in the same way as Lemma 7.4.31.

Consider the face ring Z[S(Γ)], obtained by taking quotient of the polynomialring on generators v

Gcorresponding to non-empty faces of Γ by relations (7.47).

The grading is given by deg vG= 2(n− dimG).

Example 7.9.11. Let Γ be the torus graph shown in Fig. 7.7 (b). Denoteits vertices by p and q, the edges by e, g, h, and their opposite 2-faces by E,G, H , respectively. The simplicial cell complex S(Γ) is obtained by gluing twotriangles along their boundaries. The face ring Z[S(Γ)] is the quotient of the gradedpolynomial ring

Z[vE, v

G, v

H, vp, vq], deg v

E= deg v

G= deg v

H= 2, deg vp = deg vq = 6

by the two relations

vEvGvH= vp + vq, vpvq = 0.

(The generators ve, vg, vh can be excluded using relations like ve = vGvH

.)

By definition, the equivariant cohomology of a T -graph comes together with amonomorphism into the sum of polynomial rings:

r : H∗T (Γ) −→

⊕

V (Γ)

H∗(BT ).

A similar map for the face ring Z[S(Γ)] is given by Theorem 3.5.6 (or by Theo-rem 7.4.23). The latter map can be written in our case as

s : Z[S(Γ)] −→⊕

v∈V (Γ)

Z[S(Γ)]/(vG: G 6∋ v).

Theorem 7.9.12 ([204]). The equivariant cohomology ring H∗T (Γ) of a T -graph

Γ is isomorphic to the face ring Z[S(Γ)]. In other words, H∗T (Γ) is isomorphic to

the quotient of the polynomial ring on the Thom classes τG

by relations (7.47).

Proof. We define a map

Z[vG: G is a face] −→ H∗

T (Γ)

by sending vG

в τG. By Lemma 7.9.9, it factors through a map ϕ : Z[S(Γ)]→ H∗

T (Γ).This map is surjective by Lemma 7.9.10. Also, ϕ is injective, because we haves = r ϕ and s is injective by Theorem 3.5.6.


Pseudomanifolds and orientations. Which simplicial posets arise as theposets of faces of T -graphs? Here we obtain a partial answer to this question.

The definition of pseudomanifold (Definition 4.6.4) can be extended easily tosimplicial posets:

Definition 7.9.13. A simplicial poset S of rank n is called an (n − 1)-dimensional pseudomanifold (without boundary) if

(a) for any element σ ∈ S, there is an element τ of rank n such that σ 6 τ(in other words, S is pure (n− 1)-dimensional);

(b) for any element σ ∈ S of rank (n− 1) there are exactly two elements τ ofrank n such that σ < τ ;

(c) for any two elements τ and τ ′ of rank n there is a sequence of elementsτ = τ1, τ2, . . . , τk = τ ′ such that rank τi = n and τi ∧ τi+1 contains anelement of rank (n− 1) for i = 1, . . . , k − 1.

Simplicial cell decompositions of topological manifolds are pseudomanifolds,but there are pseudomanifolds that do not arise in this way, see Example 7.9.15.

Theorem 7.9.14 ([204]).

(a) Let Γ be a torus graph; then S(Γ) is a pseudomanifold, and the face ringZ[S(Γ)] admits an lsop;

(b) Given an arbitrary pseudomanifold S and an lsop in Z[S], one can canon-ically construct a torus graph ΓS .

Furthermore, ΓS(Γ) = Γ.

Proof. (a) Vertices of S(Γ) correspond to (n−1)-faces of Γ. Since any face of Γcontains a vertex and Γ is n-valent, S(Γ) is pure (n−1)-dimensional. Condition (b)from the definition of a pseudomanifold follows from the fact that an edge of Γ hasexactly two vertices, and (c) follows from the connectivity of Γ. In order to findan lsop, we identify Z[S(Γ)] with a subset of H∗(BT )V (Γ) (see Theorem 7.9.12)and consider the constant map H∗(BT ) → H∗(BT )V (Γ). It factors through amonomorphism H∗(BT )→ Z[S(Γ)], and Lemma 3.5.8 implies that the image of abasis in H2(BT ) is an lsop.

(b) Let S be a pseudomanifold of dimension (n− 1). Define a graph ΓS whosevertices correspond to (n−1)-dimensional simplices σ ∈ S, and in which the numberof edges between two vertices σ and σ′ is equal to the number of (n−2)-dimensionalsimplices in σ ∧ σ′. Then ΓS is a connected n-valent graph, and we need to definean axial function.

We can regard an lsop as a map λ : H∗(BT ) → Z[S]. Assume that S has melements of rank 1 (we do not call them vertices to avoid confusion with the verticesof Γ) and let v1, . . . , vm be the corresponding degree-two generators of Z[S]. Thenfor t ∈ H2(BT ) we can write

λ(t) =

m∑

i=1

λi(t)ui,

where λi is a linear function on H2(BT ), that is, an element of H2(BT ). Let ebe an oriented edge of Γ with initial vertex v = i(e). Then v corresponds to an(n− 1)-simplex of S, and we denote by I(v) ⊂ 1, . . . ,m the corresponding set ofrank 1 elements of S; note that |I(v)| = n. Since λ is an lsop, the set λi : i ∈ I(v)is a basis of H2(BT ). Now we define the axial function α : E(Γ) → H2(BT ) by


requiring that its value on E(Γ)v is the dual basis of λi : i ∈ I(v). In more detail,the edge e corresponds to an (n − 2)-simplex of S and let ℓ ∈ I(v) be the uniqueelement which is not in this (n− 2)-simplex. Then we define α(e) by requiring that

(7.48) 〈α(e), λi〉 = δiℓ, i ∈ I(v),where δiℓ is the Kronecker delta. We need to check the three conditions from thedefinition of axial function. Let v′ = t(e) = i(e). Note that the intersection of I(v)and I(v′) consists of at least (n− 1) elements. If I(v) = I(v′) then Γ has only twovertices, like in Example 7.9.3, while S is obtained by gluing together two (n− 1)-simplices along their boundaries, see Example 3.6.5. Otherwise, |I(v)∩I(v′)| = n−1and we have ℓ /∈ I(v′). Let ℓ′ be an element such that ℓ′ ∈ I(v′), but ℓ′ /∈ I(v).Then (7.48) implies that 〈α(e), λi〉 = 〈α(e), λi〉 = 0 for i ∈ I(v)∩ I(v′). As we workwith integral bases, this implies α(e) = ±α(e). It also follows that α(E(Γ)v \e) andα(E(Γ)v′ \ e) give the same bases in the quotient space H2(BT )/α(e). Identifyingthese bases, we obtain a connection θe : E(Γ)v → E(Γ)v′ satisfying α(θe(e

′)) ≡ α(e′)mod α(e) for any e′ ∈ E(Γ)v, as needed.

The identity ΓS(Γ) = Γ is obvious.

Theorem 7.9.14 would have provided a complete characterisation of simplicialposets arising from T -graphs if one had S(ΓS) = S. However, this is not the casein general, as is shown by the next example:

Example 7.9.15. Let K be a triangulation of a 2-dimensional sphere differentfrom the boundary of a simplex. Choose two vertices that are not joined by an

edge. Let K be the complex obtained by identifying these two vertices. Then Kis a pseudomanifold. If Z[K] admits an lsop, then Z[K] also admits an lsop (this

follows easily from Lemma 3.5.8). However, S(ΓK) 6= K (in fact, S(ΓK) = K). It

follows that K does not arise from any T -graph.

Definition 7.9.16. A map o : V (Γ) → ±1 is called an orientation of a T -graph Γ if o(i(e))α(e) = −o(i(e))α(e) for any e ∈ E(Γ).

Example 7.9.17. LetM be a torus manifold which admits a T -invariant almostcomplex structure. The associated axial function αM satisfies αM (e) = −αM (e)for any oriented edge e. In this case we can take o(v) = 1 for every v ∈ V (ΓM ).

Proposition 7.9.18. An omniorientation of a torus manifold M induces anorientation of the associated T -graph ΓM .

Proof. For any vertex v ∈ MT = V (ΓM ) we set o(v) = σ(v), where σ(v) isthe sign of v (see Lemma 7.3.18).

Example 7.9.19. Let Γ be a complete graph on four vertices v1, v2, v3, v4.Choose a basis t1, t2, t3 ∈ H2(BT 3) and define an axial function by setting

α(v1v2) = α(v3v4) = t1, α(v1v3) = α(v2v4) = t2, α(v1v4) = α(v2v3) = t3

and α(e) = α(e) for any oriented edge e. A direct check shows that this T -graphis not orientable. This graph is associated with the pseudomanifold (simplicialcell complex) shown in Fig. 7.8 via the construction of Theorem 7.9.14 (b). Thispseudomanifold S is homeomorphic to RP 2 (the opposite outer edges are identifiedaccording to the arrows shown), the ring Z[S] has three two-dimensional generatorsvp, vq, vr, which constitute an lsop. Note that RP 2 itself is non-orientable.

It follows that this T -graph does not arise from a torus manifold.


r

r

r

rr

q p

p q

r

Figure 7.8. Simplicial cell decomposition of RP 2 with 3 vertices.

Proposition 7.9.20. A T -graph is Γ is orientable if and only if the associatedpseudomanifold S(Γ) is orientable.

Proof. Let v ∈ V (Γ) and let σ be the corresponding (n− 1)-simplex of S(Γ).The oriented edges in E(Γ)v canonically correspond to the vertices of σ. Choosea basis of H2(BT ). Assume first that S(Γ) is oriented. Choose a ‘positive’ (i.e.compatible with the orientation) order of vertices of σ; this allows us to regardα(E(Γ)v) as a basis of H2(BT ). We set o(v) = 1 if it is a positively oriented basis,and o(v) = −1 otherwise. This defines an orientation on Γ. To prove the oppositestatement we just reverse this procedure.

Blowing up T -manifolds and T -graphs. Here we elaborate on relating thefollowing three geometric constructions:

(a) blowing up a torus manifold at a facial submanifold (Construction 7.4.43);(b) truncating a simple polytope at a face (Construction 1.1.12) or, more

generally, blowing up a GKM graph or a T -graph;(c) stellar subdivision of a simplicial complex or simplicial poset (see Defini-

tion 2.7.1 and Section 3.6).

The construction of blow-up of a GKM-graph is described in [145, §2.2.1]. Italso applies to T -graphs and agrees with the topological picture for graphs arisingfrom torus manifolds.

Let (Γ, α, θ) be a T -graph and G its k-face. The blow-up of Γ at G is a T -graph

Γ which is defined as follows. Its vertex set is V (Γ) = (V (Γ) \ V (G)) ∪ V (G)n−k,that is, each vertex p ∈ V (G) is replaced by n − k new vertices p1, . . . , pn−k. Itis convenient to choose these new vertices close to p on the edges from the setEp(Γ) \Ep(G), and we denote by p′i the endpoint of the edge of Γ containing bothp and pi. Furthermore, for any two vertices p, q ∈ G joined by an edge pq we index

the corresponding new vertices of G in the way compatible with the connection, i.e.

so that θpq(pp′i) = qq′i. Now we need to define the edges of the new graph Γ and

the axial function α : E(Γ) → H∗(BT ). We have four types of edges in Γ, whichare given in the following list together with the values of the axial function:

(a) pipj for every p ∈ V (G); α(pipj) = α(pp′j)− α(pp′i);(b) piqi if p and q where joined by an edge in G; α(piqi) = α(pq);(c) pip

′i for every p ∈ V (G); α(pip

′i) = α(pp′i);

(d) edges ‘left over from Γ’, i.e. e ∈ E(Γ) with i(e) /∈ V (G) and t(e) /∈ V (G);α(e) = α(e),

see Fig. 7.9 (n = 3, k = 1) and Fig. 7.10 (n = 3, k = 0).


❯

❯

rr

❳❳❳❳❳❳③❯

❳❳❳❳❳❳③❯

r rr r

α(e1) α(e2)

α(e3)

α(e4)

α(e5)

α(e1) α(e2)

α(e2)−α(e1)α(e3)

α(e3)

α(e5)

α(e5)−α(e4)

b

Γ Γ

Figure 7.9. Blow-up at an edge

❫

r

rrr

α(e1)

α(e2)

α(e3)

α(e1)

α(e2)

α(e2)−α(e1)

α(e3)

α(e3)−α(e1)

α(e3)−α(e2)

b

Γ Γ

Figure 7.10. Blow-up at a vertex

There is a blow-down map b : Γ→ Γ taking faces to faces. The face G is blown

up to a new facet G ⊂ Γ (unless G is a facet itself, in which case Γ = Γ). For any

face H ⊂ Γ not contained in G, there is a unique face H ⊂ Γ that is blown downonto H . The blow-down map induces a homomorphism in equivariant cohomology

b∗ : H∗T (Γ)→ H∗

T (Γ), which is defined by the following commutative diagram

(7.49)

H∗T (Γ)

b∗−−−−→ H∗T (Γ)

r

yyr

H∗(BT )V (Γ) Vb∗−−−−→ H∗(BT )V (Γ),

where r and r are the monomorphisms from the definition of equivariant coho-mology of a T -graph, and V (b)∗ is the homomorphism induced by the set map

V (b) : V (Γ)→ V (Γ). The next lemma describes the images of the two-dimensionalgenerators τ

F∈ H∗

T (Γ) corresponding to facets F ⊂ Γ.

Lemma 7.9.21. For a given facet F ⊂ Γ, we have b∗(τF) = τ

G+ τ

F, if G ⊂ F

and b∗(τF) = τ

Fotherwise.

Proof. We consider diagram (7.49) and check that the images of b∗(τF) and

τG+ τ

F(or τ

F) under the map r agree. Take a vertex p ∈ V (Γ). If p /∈ G, then

b−1(p) = p and r(τF)(p) = r(τ

F)(p), r(τ

G)(p) = 0. Now let p ∈ G; then b(pi) = p

for 1 6 i 6 n− k.


First assume G 6⊂ F (see Fig. 7.11). If p /∈ F , then r(τF)(p) = r(τ

F)(pi) = 0.

Otherwise p ∈ F ∩ G. Let e be the unique edge such that e ∈ Ep(Γ) and e /∈ F .Then e = pq for some q ∈ V (G) (because G 6⊂ F ). From (7.46) we obtain

r(τF)(p) = α(pq) = α(piqi) = r(τ

F)(pi), 1 6 i 6 n− k.

Then Vb∗r(τF) = r(τ

F), and therefore, b∗(τ

F) = τ

F.

rr

❳❳❳❳❳❳

❳❳❳❳❳❳

r rr r

G

p

q

F

G

pi

qi

F

b

Figure 7.11.

Now assume G ⊂ F (see Fig. 7.12). In this case the unique edge e for whiche ∈ Ep(Γ) and e /∈ F is of type pp′j . Using (7.46) we calculate

r(τF)(p) = α(pp′j),

r(τF)(pi) = α(pipj) = α(pp′j)− α(pp′i),

r(τG)(pi) = α(pip

′i) = α(pp′i), 1 6 i 6 n− k.

Then Vb∗r(τF) = r(τ

G) + r(τ

F), and therefore, b∗(τ

F) = τ

G+ τ

F.

rr

r r

❳❳❳❳❳❳❯

❳❳❳❳❳❳②

❳❳❳❳❳❳②

r rr r

r r

G

p

p′j p′i

F

G

pi

pj

p′ip′j

F

b

Figure 7.12.

Corollary 7.9.22. After the identifications H∗T (Γ)

∼= Z[S(Γ)] and H∗T (Γ)

∼=Z[S(Γ)], the equivariant cohomology homomorphism b∗ induced by the blow-down

map b : Γ→ Γ coincides with the homomorphism β from Lemma 3.6.4.

Proof. Recall from Theorem 7.9.12 that the poset S(Γ) is formed by facesof Γ with the reversed inclusion relation, and the isomorphism H∗

T (Γ)∼= Z[S(Γ)]

is established by identifying Thom classes τH

with generators vH

corresponding tofaces H ⊂ Γ. Let σ ∈ S(Γ) be the element corresponding to the blown up face G.


Then an element τ ∈ S(Γ) belongs to stS(Γ) σ if and only if its corresponding faceH ⊂ Γ satisfies G ∩H 6= ∅. The degree-two generators v1, . . . , vm of Z[S(Γ)] and

of Z[S(Γ)] correspond to the generators τF1, . . . , τ

Fmof H∗

T (Γ) and the generators

τF1, . . . , τ

Fmof H∗

T (Γ), respectively. Making the appropriate identifications, we see

that the homomorphism from Lemma 3.6.4 is determined uniquely by the conditions

τH7→ τ

Hif G ∩H = ∅,

τFi7→ τ

G+ τ

Fiif G ⊂ Fi,

τFi7→ τ

Fiif G 6⊂ Fi.

The blow-down map b∗ satisfies these conditions, thus finishing the proof.

Exercises.

7.9.23. A connection θ satisfying condition (c) from the definition of axialfunction is unique if elements of α(E(Γ)v) are 3-independent for any vertex v.

CHAPTER 8

Homotopy theory of polyhedral products

The homotopy-theoretical study of toric spaces, such as moment-angle com-plexes or general polyhedral products, has recently evolved into a separate branchlinking toric topology to unstable homotopy theory. Like toric varieties in algebraicgeometry, polyhedral product spaces provide an effective testing ground for manyimportant homotopy-theoretical techniques.

Basic homotopical properties of polyhedral products were described in our 2002text [55]; these are included in Section 4.3 of this book. Two important develop-ments have followed shortly. First, Grbic and Theriault [132], [133] describeda wide class of simplicial complexes K whose corresponding moment-angle com-plex ZK is homotopy equivalent to a wedge of spheres. (This class includes, forexample, i-dimensional skeleta of a simplex ∆m−1 for all i and m.) Second, for-mality of the Davis–Januszkiewicz space DJ (K) (or, equivalently, the polyhedralproduct (CP∞)K) was established by Notbohm and Ray in [243]. (An alternativeproof of this result was also given in [57, Lemma 7.35].) The importance of thehomotopy-theoretical viewpoint on toric spaces has been emphasised in two pa-pers [63] and [260], both coauthored with Ray and appeared in the proceedings ofOsaka 2006 conference on toric topology. Following the earlier work of Panov, Rayand Vogt [261], in [260] categorical methods have been brought to bear on torictopology. We include the main results of [260] and [261] in Sections 8.1 and 8.4.

The idea is to exhibit a toric space as the homotopy colimit of a diagram ofspaces over the small category cat(K), whose objects are the faces of a finite sim-plicial complex K and morphisms are their inclusions. The corresponding cat(K)-diagrams can also be studied in various algebraic Quillen model categories, andtheir homotopy (co)limits can be interpreted as algebraic models for toric spaces.Such models encode many standard algebraic invariants, and their existence is as-sured by the Quillen structure. Several illustrative calculations will be provided. Inparticular, we it is proved that toric and quasitoric manifolds (and various gener-alisations) are rationally formal, and that the polyhedral product (CP∞)K (or theDavis–Januszkiewicz space) is coformal precisely when K is flag.

A number of papers on the homotopy-theoretical aspects of toric spaces hasappeared since 2008. One of the most important contributions was the 2010 workof Bahri, Bendersky, Cohen and Gitler [14] establishing a decomposition of thesuspension of a polyhedral product (X,A)K into a wedge of suspensions corre-sponding to subsets I ⊂ [m]. In particular, the moment-angle complex ZK breaksup into a wedge of suspensions of full subcomplexes KI after one suspension, pro-viding a homotopy-theoretical interpretation of the cohomology calculation of The-orem 4.5.8. We include the results of [14] and related results on stable decomposi-tions of polyhedral products in Section 8.3.

321

322 8. HOMOTOPY THEORY OF POLYHEDRAL PRODUCTS

In Section 8.4 we study the loop spaces on moment-angle complexes and poly-hedral products, by applying both categorical decompositions and the classicalhomotopy-theoretical approach via the (higher) Whitehead and Samelson products.

In the last section we restrict our attention to the case of flag complexes K.We describe the Pontryagin algebra H∗(Ω(CP∞)K) explicitly by generators andrelations in Theorem 8.5.2, and also exhibit it as a colimit (or graph product)of exterior algebras. For the commutator subalgebra H∗(ΩZK) a minimal set ofgenerators was constructed in [131]; it is included as Theorem 8.5.7. Anotherresult of [131] (Theorem 8.5.11) gives a complete characterisation of the class offlag complexes K for which the moment-angle complex ZK is homotopy equivalentto a wedge of spheres.

Background material on model categories and homotopy (co)limits is given inAppendix C, further references can be also found there.

8.1. Rational homotopy theory of polyhedral products

Here we construct several decompositions of polyhedral products into colimitsand homotopy colimits of diagrams over cat(K). We also establish formality ofpolyhedral powers XK with formal X , as well as formality of (quasi)toric mani-folds and some torus manifolds. This contrasts the situation with moment-anglecomplexes ZK = (D2, S1)K, which are not formal in general (see Section 4.9).

The indexing category for all diagrams in this section is cat(K) (simplices of afinite simplicial complex K and their inclusions) or its opposite cat(K)op. We recallthe following diagrams and their (co)limits which appeared earlier in this book:

· cat(K)op-diagram k[ · ]K in cga, see Exercise 3.1.15 and Lemma 3.5.11;its limit is the face ring k[K];· cat(K)-diagram DK(D

2, S1) in top, see (4.3); its colimit is the moment-angle complex ZK = (D2, S1)K;· cat(K)-diagram DK(X ,A) in top, see (4.7); its colimit is the polyhedral

product (X ,A)K;

We can generalise the first diagram above as follows. Given a sequence C =(C1, . . . , Cm) of commutative dg-algebras over Q, define the diagram

(8.1) DK(C ) : cat(K)op → cdga, I 7→⊗

i∈I

Ci,

by mapping a morphism I ⊂ J to the surjection⊗

i∈J Ci →⊗

i∈I Ci sending each

Ci with i /∈ I to 1. Then the diagram k[ · ]K corresponds to the case Ci = k[v], thepolynomial algebra on one generator of degree 2.

Proposition 8.1.1.

(a) The diagram DK(C) is Reedy fibrant. Therefore, there is a weak equiva-

lence limDK(C)≃−→ holimDK(C). In particular, there is a weak equiva-

lence Q[K] = limQ[ · ]K ≃−→ holimQ[ · ]K.(b) The diagram DK(X,A) is Reedy cofibrant whenever each Ai → Xi is a

cofibration (e.g. when (Xi, Ai) is a cellular pair). Under this condition,

there is a weak equivalence hocolimDK(X,A)≃−→ (X,A)K.

Proof. (a) Recall from Section C.1 that a catop(K)-diagram C is Reedy fi-brant when the canonical map C(I) → lim C|

catop(∂∆(I)) is a fibration for each

8.1. RATIONAL HOMOTOPY THEORY OF POLYHEDRAL PRODUCTS 323

I ∈ K. In our case,

DK(C )(I) =⊗

i∈I

Ci, limDK(C )|cat

op(∂∆(I)) =⊗

i∈I

Ci/I,

where I is the ideal generated by all products∏i∈I ci with ci ∈ C+

i . Hence theReedy fibrance condition is satisfied. Proposition C.3.3 implies that the canonical

map limDK(C )≃−→ holimDK(C ) is a is a weak equivalence.

(b) A cat(K)-diagram D in top is Reedy cofibrant whenever each mapcolimD|

cat(∂∆(I)) → D(I) is a cofibration. In our case,

colimDK(X ,A)|cat(∂∆(I)) = (X ,A)∂∆(I) ×A[m]\I , DK(X ,A)(I) = (X ,A)I ,

so the Reedy cofibrance condition is satisfied.

We now consider more specific models for particular polyhedral products andquasitoric manifolds. All cohomology in this section is with rational coefficients.

Formality of XK and (CP∞)K. Recall that we use the notation XK for thepolyhedral product (X , pt)K, and a spaceX is formal if the commutative dg-algebraAPL(X) = A∗(S•X) is weakly equivalent to its cohomology H∗(X).

Theorem 8.1.2. If each space Xi in X = (X1, . . . , Xm) is formal, then the

polyhedral product XK is also formal.

Proof. For notational clarity, we denote the colimit of the diagram DK(X , pt)

defining XK by colimI XI , where I ∈ K. We shall prove that APL = A∗S• maps

this colimit to the limit of dg-algebras APL(XI) ∼=

⊗i∈I APL(Xi). For this it is

convenient to work with simplicial sets, as the definition of the polynomial de Rhamfunctor A∗ implies that it takes colimits in sset to limits in cdga, see [39, §13.5].

We have a natural weak equivalence |S•(XI)| → X I . Since the total singular

complex functor S• is right adjoint, it preserves products, so we have an equivalence|(S•X )I | → X I . The cat(K)-diagram DK(X , pt) given by I 7→ X I is cofibrantby Proposition 8.1.1, and the diagram I 7→ |(S•X )I | is cofibrant by the same

reason. We therefore have a weak equivalence colimI |(S•X )I | → colimI XI by

Proposition C.3.4. Applying APL, we obtain the zigzag

APL(XK) = APL colimI X

I ≃−→ APL colimI |(S•X )I | ∼= APL| colimI(S•X )I |,where in the last identity we used the fact that the realisation functor is left adjointand therefore preserves colimits. Given an arbitrary simplicial set Y•, there is a

quasi-isomorphism APL(|Y•|) = A∗(S•|Y•|) ≃−→ A∗(Y•) induced by the equivalenceY• → S•|Y•|. We therefore can continue the zigzag above as

APL| colimI(S•X )I | ≃−→ A∗(colimI(S•X )I) ∼= limI A∗(S•X )I = limI APL(X

I).

Now, since each Xi is formal, there is a zigzag of quasi-isomorphismsAPL(Xi) ← · · · → H∗(Xi). Applying Proposition 8.1.1 (a) for the caseCi = APL(Xi) and Ci = H∗(Xi) we obtain that both the corresponding diagramsDK(C ) are fibrant, so their limits are weakly equivalent by Proposition C.3.4:

limI APL(XI)

≃←− · · · ≃−→ limI H∗(X I)

(here we also use the fact that H∗(X I) ∼=⊗

i∈I H∗(Xi), as we work with rational

coefficients). The proof is finished by appealing to the isomorphism

limI H∗(X I) ∼= H∗(XK).


In the case Xi = CP∞ this is proved in Proposition 4.3.1 (we only need this casein this section). For the general case, the proof will be given in Section 8.3.

Corollary 8.1.3. The Davis–Januszkiewicz space ETm ×Tm ZK is formal.

Proof. This follows from the homotopy equivalence (CP∞)K ≃ ETm×TmZK

of Theorem 4.3.2.

Remark. The case Xi = CP∞ of Theorem 8.1.2 (formality of the Davis–Januszkiewicz space) was proved in [243, Theorem 5.5] and [57, Lemma 7.35]. Ac-cording to [243, Theorem 4.8], the space (CP∞)K is integrally formal, i.e. the sin-gular cochain algebra C∗((CP∞)K;Z) is formal as a non-commutative dg-algebra.

The result of Theorem 8.1.2 cannot be extended to polyhedral products of theform (X ,A)K. Although limI APL((X ,A)I) is still a model for APL(X ,A)K (seethe next subsection), the cat(K)op-diagram I 7→ H∗((X ,A)I) is not fibrant ingeneral, and therefore its limit is neither isomorphic to limI APL((X ,A)I), nor toH∗((X ,A)K). Indeed, as we have seen in Section 4.9, the moment-angle complexZK = (D2, S1)K is not formal in general.

The argument in the proof of Theorem 8.1.2 shows that rational cohomology,as a functor from top to cdga, maps homotopy colimit of diagram DK(X , pt) tohomotopy limit of diagram DK(H∗(X )).

The coformality of (CP∞)K is explored in Theorem 8.5.6.

Models for (X ,A)K and ZK. Given a polyhedral product (X ,A)K =colimI(X ,A)I , we can consider the cat(K)op-diagram of commutative dg-algebrasdefined by I 7→ APL((X ,A)I), and denote its limit by limI APL((X ,A)I).

Proposition 8.1.4. There is a quasi-isomorphism of commutative dg-algebras

APL((X,A)K)≃−→ limI APL((X,A)I).

Proof. Repeat literally the argument in the first part of proof of Theo-rem 8.1.2 (before appealing to the formality of Xi).

More specific models can be obtained in the particular case ZK = (D2, S1)K.Alongside with the diagram DK(D

2, S1) we consider the cat(K)-diagramDK(pt , S

1) in top whose value on I ⊂ J is the quotient map of tori

(8.2) Tm/T I = (S1)[m]\I → (S1)[m]\J = Tm/T J .

This diagram is not cofibrant; we denote its homotopy colimit by hocolimI Tm/T I.

Proposition 8.1.5 ([261]). There is a weak equivalence ZK≃ hocolimI Tm/T I.

Proof. Objectwise projections (D2, S1)I → (S1)[m]\I induce a weak equiva-lence of diagrams DK(D

2, S1) → DK(pt , S1), whose source is Reedy cofibrant but

whose target is not. Proposition C.3.2 therefore determines a weak equivalence

ZK = colimI(D2, S1)I

≃−→ hocolimI Tm/T I .

In order to obtain a rational model of ZK from this homotopy limit decomposi-tion, we consider catop(K)-diagrams Λ[m]⊗Q[ · ]K and Λ[m]/Λ[ · ]K in cdga. Thefirst is obtained by objectwise tensoring the diagram Q[ ·]K with the exterior algebraΛ[m] = Λ[u1, . . . , um], deg ui = 1, and imposing the standard Koszul differential.So the value of Λ[m]⊗Q[ · ]K on I ⊂ J is the quotient map

(Λ[m]⊗Q[vi : i ∈ J ], d

)→(Λ[m]⊗Q[vi : i ∈ I], d

),


where d is defined on Λ[m]⊗Q[vi : i ∈ I] by dui = vi for i ∈ I and dui = 0 otherwise.The value of the second diagram Λ[m]/Λ[ · ]K on I ⊂ J is the monomorphism

Λ[ui : i /∈ J ]→ Λ[ui : i /∈ I]of algebras with zero differential. Objectwise projections induce a weak equivalence

(8.3) Λ[m]⊗Q[ · ]K → Λ[m]/Λ[ · ]K

in [catop(K),cdga] , whose source is Reedy fibrant but whose target is not.The following result describes the first algebraic model of ZK, and also recovers

the main result of Section 4.5 with rational coefficients.

Theorem 8.1.6 ([260]). The commutative differential graded algebras APL(ZK)and holimΛ[m]/Λ[ · ]K = holimI Λ[ui : i /∈ I] are weakly equivalent in cdga.

Proof. We first construct models for the products of disks and circles(D2, S1)I which are compatible with the inclusions (D2, S1)I ⊂ (D2, S1)J form-ing the diagram DK(D

2, S1). The model of a single disk D2 is the Koszul algebra(Λ[u]⊗Q[v], d). The map Λ[u]⊗Q[v]→ APL(D

2) takes u to the form ω = xdy−ydx(where (x, y) ∈ D2 are the standard cartesian coordinates), and takes v to its dif-ferential 2dx ∧ dy. It is important that the map Λ[u] ⊗ Q[v] → APL(D

2) restrictsto the quasi-isomorphism Λ[u] → APL(S

1) taking u to the form dϕ representinga generator of H1(S1). (Here ϕ is the polar angle; note that the restriction ofxdy−ydx = r2dϕ to S1 is closed, but not exact, because ϕ is not a globally definedfunction.) Taking product yields compatible quasi-isomorphisms

(Λ[u1, . . . , um]⊗Q[vi : i ∈ I], d

) ≃−→ APL((D2, S1)I),

and therefore a weak equivalence of Reedy fibrant diagrams in cdga. Their lim-its are therefore quasi-isomorphic, which together with the quasi-isomorphisms ofProposition 8.1.4, Proposition C.3.3 and (8.3) gives the required zigzag

APL(ZK) = APL((D2, S1)K

) ≃−→ limI APL((D2, S1)I

)

≃←− limI

(Λ[u1, . . . , um]⊗Q[vi : i ∈ I], d

) ≃−→ holimI Λ[ui : i /∈ I].

On the other hand, the zigzag above together with Exercise 3.1.15 (orLemma 3.5.11) gives a weak equivalence

(8.4) APL(ZK) ≃ limI

(Λ[m]⊗Q[vi : i ∈ I], d

)=(Λ[m]⊗Q[K], d

),

which implies the isomorphism of Theorem 4.5.4 in the case of rational coefficients:

H∗(ZK;Q) ∼= H(Λ[u1, . . . , um]⊗Q[K], d

).

To obtain the second algebraic model for ZK we regard it as the homotopy fibreof the inclusion (CP∞)K → (CP∞)m = BTm (see Theorem 4.3.2). We thereforecan identify ZK with the limit of the following diagram in top:

(8.5)

ETmy

(CP∞)K −−−−→ BTm

This diagram is fibrant for the appropriate Reedy structure, as ETm → BTm

is a fibration (see Proposition C.3.5 (b)), so its limit is weakly equivalent to thehomotopy limit.


Now we apply APL to (8.5), on the understanding that it does not generallyconvert pullbacks to pushouts [39, §3]. We obtain the diagram

(8.6)

APL(BTm) −−−−→ APL(ET

m)y

APL((CP∞)K)

in cdga, which is not Reedy cofibrant. Here is the second model for ZK:

Theorem 8.1.7 ([260]). The algebra APL(ZK) is weakly equivalent the homo-topy colimit of (8.6) in cdga.

Proof. There is an objectwise weak equivalence mapping the diagram

Q[v1, . . . , vm] −−−−→(Λ[u1, . . . , um]⊗Q[v1, . . . , vm], d

)y

Q[K]to (8.6); here the upper arrowQ[m]→ (Λ[m]⊗Q[m], d) is the standard model for thefibration ETm → BTm, and Q[K] is a model for APL((CP∞)K) by Theorem 8.1.2.The above diagram is cofibrant, because the upper arrow is a cofibration in cdga,and its colimit is (Λ[m]⊗Q[K], d

). So the result follows from (8.4).

Theorem 8.1.7 chimes with the rational models of fibrations from [112, §15(c)].

Remark. We may summarise the results of Theorems 8.1.6 and 8.1.7 as fol-lows. As functors top→ cdga, both rational cohomology and APL map homotopycolimits to homotopy limits on diagrams DK(pt , S

1), I 7→ Tm/T I , and map homo-topy limits to homotopy colimits on diagrams (8.5).

Models for toric and quasitoric manifolds. Let M = M(P,Λ) be a qua-sitoric manifold over a simple n-polytope P with characteristic map Λ : Zm → Zn.Here we denote byK the dual sphere triangulationKP of P . We recall from Proposi-tion 7.3.12 that M can be identified with the quotient of the moment-angle manifoldZP = ZK by the free action of the (m − n)-torus K(Λ) = Ker(Λ : Tm → T n); wedenote the map of tori defined by Λ by the same letter for simplicity. All resultsbelow are equally applicable to toric manifolds M , in which case K is the underlyingcomplex of the corresponding complete regular simplicial fan.

Our topological and algebraic models of M are obtained from those of ZK bythe appropriate factorisation by the action of K(Λ).

We consider the cat(K)-diagram DK(pt , S1)/K(Λ) obtained by factorisation

of (8.2); its value on I ⊂ J is the quotient map of tori

T n/Λ(T I)→ T n/Λ(T J),

where we identified Tm/K(Λ) with T n. This diagram is not cofibrant; we denoteits homotopy colimit by hocolimI T

n/Λ(T I).The following result was first proved by Welker, Ziegler and Zivaljevic in [321].

It appears to be the earliest mention of homotopy colimits in the toric context, andrefers to a diagram that is clearly not Reedy cofibrant:

Proposition 8.1.8. There is a weak equivalence M ≃ hocolimI Tn/Λ(T I).


Proof. As in the proof of Proposition 8.1.5, we consider objectwise projections(D2, S1)I → Tm/T I = (S1)[m]\I , and take quotients by the action of K(Λ). As aresult, we obtain a weak equivalence

M = ZK/K(Λ) = colimI

((D2, S1)I/K(Λ)

) ≃−→ hocolimI Tn/Λ(T I).

Next we construct an analogue of the algebraic model (8.4) for quasitoric man-ifolds. For this we consider the elements

ti = λi1v1 + · · ·+ λimvm, 1 6 i 6 n,

in the face ring Q[K] = Q[v1, . . . , vm]/IK corresponding to the rows of matrix Λ.

Lemma 8.1.9. For a toric or quasitoric manifold M , the algebra APL(M) isweakly equivalent to the commutative dg-algebra

(Λ[x1, . . . , xn]⊗Q[K], d

), with dxi = ti, dvi = 0.

Proof. The argument is similar to that for Theorem 8.1.6. We consider acatop(K)-diagram Λ[n]⊗Q[ · ]K, whose value on I ⊂ J is the quotient map

(Λ[x1, . . . , xn]⊗Q[vi : i ∈ J ], d

)→(Λ[x1, . . . , xn]⊗Q[vi : i ∈ I], d

),

where d is defined on Λ[n]⊗Q[vi : i ∈ I] by dxi = ti for i ∈ I and dxi = 0 otherwise.Then define quasi-isomorphisms

(Λ[x1, . . . , xn]⊗Q[vi : i ∈ I], d

) ≃−→ APL((D2, S1)I/K(Λ)

)

by sending each xi to the K(Λ)-invariant 1-form λi1r21dϕ1+ · · ·+λimr2mdϕm, where

(ri, ϕi) are polar coordinates on the ith disk or circle. These quasi-isomorphismsare compatible with the maps corresponding to inclusions of simplices I ⊂ J andtherefore provide a weak equivalence of Reedy fibrant diagrams in cdga. Theirlimits are therefore quasi-isomorphic, and we obtain the required zigzag

APL(M) = APL((D2, S1)K/K(Λ)

) ≃−→ limI APL((D2, S1)I/K(Λ)

)

≃←− limI

(Λ[x1, . . . , xn]⊗Q[vi : i ∈ I], d

)=(Λ[x1, . . . , xn]⊗Q[K], d

).

Remark. The (quasi)toric manifold M = M(P,Λ) is the homotopy fibre ofthe composition of the inclusion (CP∞)K → BTm and the map BΛ : BTm → BT n

(an exercise). In other words, there is a homotopy pullback diagram

(8.7) M //

ET n

(CP∞)K

i // BTmBΛ // BT n

The model of Lemma 8.1.9 can be obtained by applying the results of [112, §15(c)]to the fibration M → (CP∞)K above.

Now we can prove the main result of this subsection:

Theorem 8.1.10 ([260]). Every toric or quasitoric manifold is formal.

Proof. We use the model of Lemma 8.1.9 and utilise the fact that Q[K] isCohen–Macaulay (Corollary 3.3.13), i.e. Q[K] is free as module over Q[t1, . . . , tn].


Hence ⊗Q[t1,...,tn]Q[K] is a right exact functor, and applying it to the quasi-isomorphism (Λ[u1, . . . , un]⊗Q[t1, . . . , tn], d)→ Q yields a quasi-isomorphism

(Λ[u1, . . . , un]⊗Q[K], d)→ Q[K]/(t1, . . . , tn),

which is given by the projection onto the second factor. Since Q[K]/(t1, . . . , tn) ∼=H∗(M) by Theorem 7.3.27, the result follows from Lemma 8.1.9.

Similar arguments apply more generally to torus manifolds over homology poly-topes (see Section 7.4), and even to arbitrary torus manifolds with zero odd dimen-sional cohomology. In the latter case, Q[K] is replaced by the face ring Q[S] of anappropriate simplicial poset S (see exercises below).

Note also that the formality of projective toric manifolds follows immediatelyfrom the fact that they are Kahler.

Exercises.

8.1.11. Construct the homotopy pullback diagram (8.7).

8.1.12. Extend the argument of Theorem 8.1.2 to show that the polyhedralpower (CP∞, pt)S (see Construction 4.10.1) is formal for any simplicial poset S.

8.1.13. Show that a torus manifoldM withHodd(M ;Z) = 0 is formal. (Hint useTheorem 7.4.35 establishing an isomorphism H∗(M) ∼= Q[S], where Q[S] is the facering of the simplicial poset dual to the quotient M/T = Q, and use Lemma 7.4.34to show that Q[S] is free over Q[t1, . . . , tn].)

8.2. Wedges of spheres and connected sums of sphere products

There are two situations when the homotopy type of the moment-angle complexZK can be described explicitly. The first concerns a family of examples of polytopalsphere triangulations K for which ZK is homeomorphic to a connected sum ofsphere products, with two spheres in each product. The proofs use differentialtopology and surgery theory; we included sample results as Theorem 4.6.12 andTheorem 6.2.10 and refer to a more detailed account in the work of Gitler andLopez de Medrano [127]. The second situation is when ZK is homotopy equivalentto a wedge of spheres; the corresponding families of examples of K were constructedby Grbic and Theriault in [132], [133] and are reviewed here.

Theorem 8.2.1 ([133]). Let K = K1 ∪I K2 be a simplicial complex obtained bygluing K1 and K2 along a common face, which may be empty. If ZK1 and ZK2 arehomotopy equivalent to wedges of spheres, then ZK is also homotopy equivalent toa wedge of spheres.

We reproduce the proof from [133], which uses two lemmata.

8.2. WEDGES OF SPHERES AND CONNECTED SUMS OF SPHERE PRODUCTS 329

Lemma 8.2.2 (Cube Lemma). Suppose there is a homotopy commutative dia-gram of spaces

E //

F

G //

H

A

// B

C // D.

Suppose the bottom face is a homotopy pushout and the four sides are homotopypullbacks. Then the top face is a homotopy pushout.

Proof. See [212, Theorem 25]. Note that the statement is fairly straight-forward in the case when D = B ∪A C and the vertical arrows are locally trivialfibre bundles obtained by pulling back the bundle H → D along the arrows of thebottom face. This will be enough for our purposes.

The join of spaces A, B is defined as the identification space

A ∗B = A×B × I / (a, b1, 0) ∼ (a, b2, 0), (a1, b, 1) ∼ (a2, b, 1).

The product A×B embeds into the join as A×B× 12 . Furthermore, the lower half

A ∗B6 12= (a, b, t) ∈ A ∗B : t 6 1

2,

of the join is the mapping cylinder of the first projection π1 : A×B → A, and theupper half A ∗B> 1

2is the mapping cylinder of the second projection π2 : A×B → B.

It follows that there is a homotopy pushout diagram

(8.8)

A×B π2 //

π1

B

A // A ∗B,

which can be viewed as the homotopy-theoretic definition of the join.For pointed spaces A, B, there are canonical homotopy equivalences ΣA∧B ≃

Σ(A∧B) ≃ A ∗B, where A∧B = (A×B)/(A×pt ∪pt ×B) is the smash product.The left half-smash product is A ⋉ B = A × B/(A × pt) and the right half-smashproduct is A⋊B = A×B/(pt ×B). Let ǫA denote the map collapsing A to a point.

Lemma 8.2.3. Let A, B, C and D be spaces. Define Q as the homotopy pushout

A×B ǫA×idB //

idA×ǫB

C ×B

A×D // Q.

Then Q ≃ (A ∗B) ∨ (C ⋊B) ∨ (A⋉D).


Proof. We can decompose the pushout square above as

(8.9)

A×B π2 //

π1

Bi2 //

C ×B

A //

i1

A ∗B j2 //

j1

E

A×D // F // Q,

where i1 and i2 denote the inclusions into the first and second factor, and eachsmall square is a homotopy pushout.

Since the map A → A ∗B is null homotopic, we can pinch out A in the leftbottom square above to obtain a homotopy pushout

pt //

A ∗Bj1

A⋉D // F.

Hence F ≃ (A ∗B) ∨ (A ⋉ D) and j1 is homotopic to the inclusion into the firstwedge summand. Similarly, E ≃ (A ∗B) ∨ (C ⋊ B). The required decompositionof Q now follows by considering the right bottom square in (8.9).

Proof of Theorem 8.2.1. Recall from Theorem 4.3.2 that ZK is the homo-topy fibre of the canonical inclusion i : (CP∞)K → (CP∞)m = BTm. Assumethat K1 is a simplicial complex on [m1], K2 is a simplicial complex on [m2] andK is a simplicial complex on [m], so that m = m1 + m2 − |I|. By some abuseof notation we assume that the set [m1] is included as the first m1 elements in[m], and [m2] is included as the last m2 elements. This defines the inclusions like(CP∞)m1 → (CP∞)m, (CP∞)K2 → (CP∞)K, etc. We have a pushout square

(CP∞)I //

(CP∞)K1

(CP∞)K2 // (CP∞)K.

Map each of the four corners of this pushout into (CP∞)m and take homotopyfibres. This gives homotopy fibrations

Tm−m2 × Tm−m1 = Tm−|I| → (CP∞)I → (CP∞)m,

ZK1 × Tm−m1 → (CP∞)K1 → (CP∞)m,

Tm−m2 ×ZK2 → (CP∞)K2 → (CP∞)m,

ZK → (CP∞)K → (CP∞)m.

Including (CP∞)I into (CP∞)K1 gives a homotopy pullback diagram

ΩBTm // Tm−|I| //

θ

(CP∞)I //

BTm

ΩBTm // ZK1 × Tm−m1 // (CP∞)K1 // BTm

8.2. WEDGES OF SPHERES AND CONNECTED SUMS OF SPHERE PRODUCTS 331

for some map θ of fibres. We now identify θ. With BTm =∏mi=1 CP

∞, the pullbackjust described is the product of the homotopy pullback

ΩBTm1 // Tm−m2 //

θ′

(CP∞)I //

BTm1

ΩBTm1 // ZK1// (CP∞)K1 // BTm1

and the path-loop fibration Tm−m1 → pt → BTm−m1 . So θ = θ′ × idTm−m1 .Further, Tm−m2 is a retract of ΩBTm1 ≃ Tm1 and ΩBTm1 → ZK1 is null homo-topic since ΩBTm1 is a retract of Ω((CP∞)K1). Hence θ′ ≃ ǫTm−m2 and thereforeθ ≃ ǫTm−m2 × idTm−m1 . A similar argument for the inclusion of (CP∞)I into(CP∞)K2 shows that the map of fibres Tm−m2 × Tm−m1 → Tm−m2 × ZK2 is ho-motopic to idTm−m2 × ǫTm−m1 .

Collecting all this information about homotopy fibres, Lemma 8.2.2 shows thatthere is a homotopy pushout

Tm−m2 × Tm−m1ǫ× id //

id×ǫ

ZK1 × Tm−m1

Tm−m2 ×ZK2

// ZK.

Lemma 8.2.3 then gives a homotopy decomposition

(8.10) ZK ≃ (Tm−m2 ∗Tm−m1) ∨ (ZK1 ⋊ Tm−m1) ∨ (Tm−m2 ⋉ ZK2).

To show ZK is homotopy equivalent to a wedge of spheres, we show that eachof Tm−m2 ∗Tm−m1, ZK1 ⋊ Tm−m1 and Tm−m2 ⋉ ZK2 is homotopy equivalent toa wedge of spheres. First, observe that the suspension of a product of spheresis homotopy equivalent to a wedge of spheres, so Tm−m2 ∗Tm−m1 is homotopyequivalent to a wedge of spheres. Second, as ZK1 is homotopy equivalent to awedge of spheres we can write ZK1 ≃ ΣW , where W is a wedge of spheres (notethat ZK1 is 2-connected by Proposition 4.3.5 (a)). We then have ZK1 ⋊ Tm−m1 ≃ΣW ⋊Tm−m1 ≃ ΣW ∨ (ΣTm−m1 ∧W ). Now ΣTm−m1 is homotopy equivalent toa wedge of spheres. Therefore, as W is homotopy equivalent to a wedge of spheresso is ΣTm−m1 ∧W . Hence ZK1 ⋊ Tm−m1 is homotopy equivalent to a wedge ofspheres. The decomposition of the summand Tm−m2 ⋉ZK2 into a wedge of spheresis exactly as for ZK1 ⋊ Tm−m1.

Corollary 8.2.4. Assume that there is an order I1, . . . , Is of the maximalfaces of K such that

(⋃j<k Ij

)∩Ik is a single face for each k = 1, . . . , s. Then ZK

has homotopy type of a wedge of spheres.

As an application, we describe the homotopy type of ZK for two particularseries of K: 0-dimensional complexes and trees (connected graphs without cycles).

Proposition 8.2.5. Let K be m disjoint points, m > 2. Then

(8.11) ZK ≃m∨

k=2

(Sk+1

)∨(k−1)(mk

).


Proof. Let Km denote the complex consisting of m disjoint points. Apply-ing (8.10) to the decomposition Km = Km−1 ⊔ K1 we obtain

(8.12) ZKm≃ (Tm−1 ∗T 1) ∨ (ZKm−1 ⋊ T 1)

(the third wedge summand vanishes because ZK1 ≃ pt). An inductive argumentusing the decomposition Σ(A×B) ≃ ΣA ∨ΣB ∨ (ΣA ∧B) shows that

Tm−1 ∗T 1 ≃ ΣΣTm−1 ≃m∨

k=2

(Sk+1

)∨(m−1k−1

).

Assuming by induction that (8.11) holds for Km−1, we obtain

ZKm−1⋊T1 ≃ ZKm−1∨ΣZKm−1 ≃

m−1∨

k=2

(Sk+1

)∨(k−1)(m−1k

)∨m∨

k=3

(Sk+1

)∨(k−2)(m−1k−1

).

Substituting the last two formulae into (8.12) we finally obtain (8.11).

Observe that ZK corresponding to m disjoint points is the homotopy fibre ofthe inclusion of the m-fold wedge (CP∞)∨m into the m-fold product (CP∞)m. Aswe have seen in Example 4.7.6, ZKm

is homotopy equivalent to the complement ofthe union of all coordinate planes of codimension two in Cm. In this context theresult of Proposition 8.2.5 was obtained in [132].

The homotopy type of ZK corresponding to a tree with m+1 vertices dependsonly on the number of vertices and does not depend of the form of the tree; also,the homotopy type of ZK corresponding to a tree with m + 1 vertices is the sameas that of ZK corresponding to m disjoint points:

Proposition 8.2.6. Let K be a tree with m+ 1 vertices, m > 2. Then

ZK ≃m∨

k=2

(Sk+1

)∨(k−1)(mk

).

Proof. This time we use the decomposition Km = Km−1 ∪v K1, where Kmdenotes a tree with m+1 vertices (so that K1 is a segment), and the union in takenalong a common vertex v. The rest of the proof is as for Proposition 8.2.5.

One can notice the similarity between the wedge decomposition of Proposi-tion 8.2.6 and the connected sum decomposition of Theorem 4.6.12. The nature ofthis similarity is explained in the work of Theriault [304].

A simplicial complex K is shifted if there is an ordering of its vertices such thatwhenever I ∈ K, i ∈ I and i < j, then (I \ i) ∪ j ∈ K.

Theorem 8.2.7 ([133, Theorem 9.4]). If K is a shifted complex, then ZK ishomotopy equivalent to a wedge of spheres.

Idea of proof. For any simplicial complex K on [m] there is a pushout square

lkmK //

K1,...,m−1

stmK // K,

where K1,...,m−1 denotes the restriction of K to the first m− 1 vertices. It gives

rise to a pushout square of the corresponding polyhedral products (CP∞)K and,

8.3. STABLE DECOMPOSITIONS OF POLYHEDRAL PRODUCTS 333

by application of Lemma 8.2.2, to a pushout square of the moment-angle com-plexes ZK. The key observation is that if K is shifted, then all three subcomplexeslkmK, stmK and K1,...,m−1 are also shifted, with respect to the induced or-dering of vertices. This allows us to use induction, in a way similar to the proof ofTheorem 8.2.1.

The i-dimensional skeleton of a simplex ∆m−1 is a shifted complex. In thiscase the dimensions of spheres in the wedge decomposition of ZK can be describedexplicitly; this result was given as Theorem 4.7.7. It also follows from a result ofPorter [264] for general polyhedral products, which we state in the next section.

Not all complexes K obtained by iterative gluing along a common face areshifted (see Exercise 8.2.8), and not all shifted complexes can be obtained by itera-tive gluing along a common face. So one can obtain even a wider class of simplicialcomplexes K whose corresponding ZK are wedges of spheres by combining the re-sults of Theorem 8.2.1 and Theorem 8.2.7.

Exercises.

8.2.8. The tree r r r is a shifted complex, but r r r r is not.

8.2.9. Let K be the graphr rr r

. Describe the homotopy type of ZK.

8.2.10. Let K be a complex obtained by iteration of the operation of attachinga k-simplex along a common (k− 1)-face, starting from a k-simplex (so K is a treewhen k = 1). Describe the homotopy type of ZK.

8.3. Stable decompositions of polyhedral products

Several important results on wedge decomposition of polyhedral products af-ter one suspension were obtained in the work of Bahri, Bendersky, Cohen andGitler [14]. These can be seen as far-reaching generalisations of the classical de-composition Σ(A × B) ≃ ΣA ∨ ΣB ∨ (ΣA ∧ B). The proofs given below arereproduced from [14] with few or no modifications. Homotopy theory of polyhe-dral products has become quite an active area, and we also review some recentresults on stable and unstable decompositions in the end of this section.

We start by defining the smash version of the polyhedral product.

Construction 8.3.1 (polyhedral smash product). The initial setup is againa simplicial complex K on [m] and a sequence of m pairs of pointed cell complexes

(X ,A) = (X1, A1), . . . , (Xm, Am).We denote the m-fold smash product of the Xi by

X∧m = X1 ∧X2 ∧ · · · ∧Xm.

Then the polyhedral smash product (X ,A)∧K is defined as the image of (X ,A)K

under the projection Xm → X∧m. More specifically, for each I ⊂ [m] we set

(X ,A)∧I =(x1, . . . , xm) ∈ X1 ∧X2 ∧ · · · ∧Xm : xj ∈ Aj for j /∈ I

,

then

(X ,A)∧K =⋃

I∈K

(X ,A)∧I =⋃

I∈K

(∧

i∈I

Xi ∧∧

i/∈I

Ai

).


Using the categorical language, define the cat(K)-diagram

(8.13)DK(X ,A) : cat(K) −→ top,

I 7−→ (X ,A)∧I ,

which maps the morphism I ⊂ J to the inclusion (X ,A)∧I ⊂ (X ,A)∧J . Then

(X ,A)∧K = colim DK(X ,A) = colimI∈K

(X ,A)∧I .

In the case when all the pairs (Xi, Ai) are the same, i.e. Xi = X and Ai = A,we use the notation (X,A)∧K for (X ,A)∧K. Also, if each Ai = pt , then we use the

abbreviated notation X∧K for (X , pt)∧K, and X∧K for (X, pt)∧K.

An inductive argument using the decomposition Σ(A×B) ≃ ΣA∨ΣB∨(ΣA∧B) shows that there is a natural pointed homotopy equivalence

(8.14) Σ(X1 × · · · × Xm)≃−→ Σ

( ∨

J⊂[m]

X∧J).

For each J ⊂ [m], define the subfamily

(X J ,AJ) = (Xj , Aj) : j ∈ J.The first result shows that the polyhedral product splits after one suspension intoa wedge of polyhedral smash products corresponding to all full subcomplexes of K:

Theorem 8.3.2 ([14]). For any sequence (X,A) of pairs of pointed cell com-plexes, homotopy equivalence (8.14) induces a natural pointed homotopy equivalence

Σ(X,A)K≃−→ Σ

( ∨

J⊂[m]

(XJ ,AJ)∧KJ

).

Proof. We have (X ,A)K = colimDK(X ,A) = colimI∈K(X ,A)I , whereDK(X ,A) is diagram (4.7). Define another diagram

EK(X ,A) : cat(K) −→ top,

I 7−→∨

J⊂[m]

(X J ,AJ )∧(I∩J).

By (8.14), there is a natural pointed homotopy equivalence

Σ(X ,A)I≃−→ Σ

( ∨

J⊂[m]

(X J ,AJ)∧(I∩J)

)

The diagrams DK(X ,A), EK(X ,A), as well as ΣDK(X ,A), ΣEK(X ,A), are obvi-ously cofibrant, so the objectwise homotopy equivalence above induces a homotopyequivalence of their colimits (see Proposition C.3.4). It remains to note that

colimΣDK(X ,A) = Σ colimDK(X ,A) = Σ(X ,A)K,

colimΣEK(X ,A) = colimI∈KΣ( ∨

J⊂[m]

(X J ,AJ)∧(I∩J)

)

= Σ( ∨

J⊂[m]

colim(I∩J)∈KJ(X J ,AJ)

∧(I∩J))

= Σ( ∨

J⊂[m]

(X J ,AJ )∧KJ

).


The homotopy type of the wedge summands (X J ,AJ)∧KJ can be described

explicitly in the case when the inclusions Ak → Xk are null-homotopic:

Theorem 8.3.3 ([14]). Let K be a simplicial complex on [m], and let (X,A) bea sequence of pairs of cell complexes with the property that the inclusion Ak → Xk

is null-homotopic for all k. Then there is a homotopy equivalence

(X,A)∧K ≃−→∨

I∈K

| lkK I| ∗ (X,A)∧I ,

where | lkK I| is the geometric realisation of the link of I in K.

Proof. By hypothesis, there is a homotopy Fk : Ak × I → Xk such thatFk(a, 0) = ik(a) and Fk(a, 1) = pt , where ik : Ak → Xk is the inclusion. By

the homotopy extension property, there exists Fk : Xk × I→ Xk with Fk(x, 0) = x,

Fk(x, 1) = gk(x) where gk : Xk → Xk is a map such that gk(a) = pt for all a ∈ Ak.Hence there is a commutative diagram

Akid //

ik

Ak

ǫ

Xk

gk // Xk

where ǫ : Ak → Xk is the constant map to the basepoint. Along with the diagram

DK = DK(X ,A) given by (8.13), define a new diagram

EK : cat(K) −→ top, I 7−→ (X ,A)∧I ,

which maps the non-identity morphism I ⊂ J to the constant map (X ,A)∧I →(X ,A)∧J to the basepoint. For every I ∈ K, define

α(I) : DK(I)→ EK(I)by α(I) = α1(I) ∧ · · · ∧ αm(I) where

αk(I) =

gk : Xk → Xk if k ∈ I,id : Ak → Ak if k /∈ I.

Since the gk are homotopy equivalences, so is α(I) for all I ∈ K. Furthermore, ifI ⊂ J , the following diagram commutes:

DK(I)α(I) //

EK(I)

DK(J)

α(J) // EK(J).

Hence the maps α(I) give a weak equivalence of diagrams DK → EK, which gives ahomotopy equivalence

hocolim DK≃−→ hocolim EK.

Finally, the diagram EK satisfies the conditions of Lemma C.3.6, so we get a homo-topy equivalence

hocolim EK ≃−→∨

I∈K

| lkK I| ∗ (X ,A)∧I


(upper semi-intervals in the face poset of K are links). The result follows since

hocolim DK = (X ,A)∧K.

Two special cases of Theorem 8.3.3 are presented next where either Ai arecontractible for all i or Xi are contractible for all i.

Theorem 8.3.4 ([14]). Let K be a simplicial complex on [m], and let (X,A) bea sequence of pairs of cell complexes with the property that all the Ai are contractible.Then there is a homotopy equivalence


(∨

I∈K

X∧I).

Proof. When all the Ai are contractible, the space (X ,A)∧I is also con-tractible unless I = [m]. By Theorem 8.3.3,

(X J ,AJ)∧KJ ≃

∨

I∈KJ

| lkKJI| ∗ (X J ,AJ)

∧I ,

which is contractible unless J ∈ KJ , i.e. J ∈ K. In the latter case we have(X J ,AJ)

∧KJ = X∧J . By Theorem 8.3.2,

Σ(X ,A)K ≃ Σ( ∨

J⊂[m]

(X J ,AJ )∧KJ

)= Σ

( ∨

J∈K

X∧J).

Remark. An interesting corollary of Theorem 8.3.4 is that the polyhedralproducts XK = (X, pt)K corresponding to simplicial complexes K with the samef -vectors become homotopy equivalent after one suspension.

Theorem 8.3.5 ([14]). Let K be a simplicial complex on [m], and let (X,A)be a sequence of pairs of cell complexes with the property that all the Xi are con-tractible. Then there is a homotopy equivalence


( ∨

J /∈K

|KJ | ∗A∧J).

Proof. Since all the Xi are contractible, all of the spaces (X ,A)∧I are alsocontractible with the possible exception of (X ,A)∧∅ = A∧m. By Theorem 8.3.3,

(X J ,AJ)∧KJ ≃

∨

I∈KJ

| lkKJI| ∗ (X J ,AJ)

∧I = | lkKJ∅| ∗ (X J ,AJ)

∧∅ = |KJ | ∗A∧J

which is contractible if J ∈ K. By Theorem 8.3.2,

Σ(X ,A)K ≃ Σ( ∨

J⊂[m]

(X J ,AJ)∧KJ

)= Σ

( ∨

J /∈K

|KJ | ∗A∧J).


Corollary 8.3.6.

(a) Let (X,A) = (D1, S0), so that (X,A)K is the real moment-angle com-plex RK. Then there is a homotopy equivalence

ΣRK≃−→

∨

J /∈K

Σ2|KJ |.

(b) Let (X,A) = (D2, S1), so that (X,A)K is the moment-angle complex ZK.Then there is a homotopy equivalence

ΣZK≃−→

∨

J /∈K

Σ2+|J||KJ |.

The above decomposition of ΣZK implies the additive isomorphism

Hk(ZK;Z) ∼=⊕

J⊂[m]

Hk−|J|−1(KJ ;Z)

of Theorem 4.5.8. Similarly, the decomposition of ΣRK implies the isomorphism

Hk(RK;Z) ∼=⊕

J⊂[m]

Hk−1(KJ ;Z).

Another result of [14] describes the cohomology ring of a polyhedral product

XK and generalises the isomorphism H∗((CP∞)K;Z) ∼= Z[K] of Proposition 4.3.1:

Theorem 8.3.7 ([14]). Let X = (X1, . . . , Xm) be a sequence of pointed cellcomplexes, and let k be a ring such that the natural map

H∗(Xj1 ;k)⊗ · · · ⊗H∗(Xjk ;k)→ H∗(Xj1 × · · · ×Xjk ;k)

is an isomorphism for any j1, . . . , jk ⊂ [m]. There is an isomorphism of algebras

H∗(XK;k) ∼=(H∗(X1;k)⊗ · · · ⊗H∗(Xm;k)

)/I,

where I is the generalised Stanley–Reisner ideal, generated by elements xj1⊗· · ·⊗xjkfor which xji ∈ H∗(Xji ;k) and j1, . . . , jk /∈ K. Furthermore, the inclusion XK →X1 × · · · ×Xm induces the quotient projection in cohomology.

Proof. By Theorem 8.3.4, there are homotopy equivalences

Σ(X1 × · · · × Xm)≃−→ Σ

( ∨

J⊂[m]

X∧J), ΣXK ≃−→ Σ

( ∨

J∈K

X∧J).

Naturality implies that the map ΣXK → Σ(X1 × · · · × Xm) is split with cofibre

Σ(∨

J /∈K X∧J). Hence there is a split cofibration

Σ( ∨

J∈K

X∧J)→ Σ

( ∨

J⊂[m]

X∧J)→ Σ

( ∨

J /∈K

X∧J).

Under the given condition on k there is a ring isomorphism

H∗(X1 × · · · ×Xm;k) ∼=⊕

J⊂[m]

H∗(Xj1 ;k)⊗ · · · ⊗ H∗(Xjk ;k).

The natural inclusion map ΣXK → Σ(X1 × · · · ×Xm) induces a map

H∗(X1 × · · · ×Xm;k)→ H∗(XK;k),


which corresponds to the projection map⊕

J⊂[m]

H∗(Xj1 ;k) ⊗ · · · ⊗ H∗(Xjk ;k)→⊕

J∈K

H∗(Xj1 ;k)⊗ · · · ⊗ H∗(Xjk ;k).

Its kernel is exactly⊕

J /∈K

H∗(Xj1 ;k)⊗ · · · ⊗ H∗(Xjk ;k)

which is the generalised Stanley–Reisner ideal I by inspection.

As in the case of the face ring k[K] = H∗((CP∞)K;k), which can be decom-posed as the limit of the cat(K)op-diagram of polynomial algebras k[vi : i ∈ I] =H∗((CP∞)I ;k), the isomorphism of Theorem 8.3.7 can be interpreted as

H∗(XK;k) ∼= limI∈KH∗(X I ;k).

This isomorphism was used in the proof of Theorem 8.1.2.There are several important situations when the isomorphism of Theorem 8.3.5

can be desuspended. As we have seen in the previous section, this is the case for thepairs (X ,A) = (D2, S1) when K is obtained by iterative gluing along a commonface or when K is shifted complex. More generally, the following result, conjecturedin [14], was proved independently by Grbic–Theriault and Iriye–Kishimoto:

Theorem 8.3.8 ([134, Theorem 1.1], [168, Theorem 1.7]). Let K be a shiftedcomplex. Let A = (A1, . . . , Am) be a sequence of pointed cell complexes, and letconeA denote the cone on A. Then there is a homotopy equivalence

(coneA,A)K≃−→

∨

J /∈K

|KJ | ∗A∧J .

Here is a case where the wedge summands can be described very explicitly:

Corollary 8.3.9. Let Ki be the i-dimensional skeleton of the simplex ∆m−1.Then there is a homotopy equivalence

(coneA,A)Ki≃−→

m∨

k=i+2

( ∨

16j1<···<jk6m

(Σi+1Aj1 ∧ · · · ∧ Ajk

)∨(k−1i+1)

).

The proof of this corollary is left as an exercise. In the case when each Ai isa loop space, Ai = ΩBi, the space (coneΩB , ΩB)Ki is the homotopy fibre of the

inclusion BKi → Bm (see Example 4.2.6.5 and Exercise 4.3.10) and decompositionabove was obtained by Porter [264] . In the case Ai = S1, Corollary 8.3.9 turnsinto Theorem 4.7.7.

The wedge decomposition of Theorem 8.3.5 can be used to describe the ringstructure for the cohomology of (X ,A)K, see [16].

Exercises.

8.3.10. Deduce Corollary 8.3.9 from Theorem 8.3.8.

8.4. Loop spaces, Whitehead and Samelson products

We now turn our attention to topological and algebraic models for the loopspaces Ω(CP∞)K and ΩZK. We can view the latter as objects in the categorytmon of topological monoids by considering Moore loops (of arbitrary length),whose composition is strictly associative.

8.4. LOOP SPACES, WHITEHEAD AND SAMELSON PRODUCTS 339

Pontryagin algebras, Whitehead and Samelson products. We loop thefibration ZK → (CP∞)K → (CP∞)m to obtain a fibration

(8.15) ΩZK −→ Ω(CP∞)K −→ Tm.

It admits a section, defined by the m generators of π2((CP∞)K) ∼= Zm, and there-fore splits in top. So we have a homotopy equivalence

Ω(CP∞)K≃−→ ΩZK × Tm

which does not preserve monoid structures.

Proposition 8.4.1. There is an exact sequence of homotopy Lie algebras

0 −→ π∗(ΩZK)⊗Q −→ π∗(Ω(CP∞)K)⊗Q −→ CL(u1, . . . , um) −→ 0,

where CL(u1, . . . , um) denotes the commutative Lie algebra with generators ui,deg ui = 1, and an exact sequence of Pontryagin algebras

(8.16) 0 −→ H∗(ΩZK;k) −→ H∗

(Ω(CP∞)K;k

)−→ Λ[u1, . . . , um] −→ 0,

for any commutative ring k with unit.

Proof. The first exact sequence follows by considering the homotopy exactsequence of the fibration (8.15), whose connecting homomorphism is zero becausethe fibration is trivial.

Since H∗(Tm;k) = Λ[u1, . . . , um] is a finitely generated free k-module, the

Kunneth formula gives an isomorphism of k-modules

H∗

(Ω(CP∞)K;k

) ∼= H∗(ΩZK;k)⊗ Λ[u1, . . . , um]

and therefore an exact sequence of k-algebras (8.16).

The homotopy group π2((CP∞)K) ∼= Zm has m canonical generators repre-sented by the maps

µi : S2 −→ CP∞ −→ (CP∞)∨m −→ (CP∞)K

for 1 6 i 6 m, where the left map is the inclusion of the bottom cell, the middlemap is the inclusion of the ith wedge summand, and the right map is the canonicalinclusion of polyhedral powers corresponding to the inclusion of the discretem-pointcomplex into K. Let

µi : S1 −→ Ω(CP∞)K

be the adjoint of µi, and let ui denote the Hurewicz image of µi in H1(Ω(CP∞)K).We shall be interested in elements of π∗(Ω(CP∞)K) represented by Samelson

products of the µi (see Section B.1 for the definition).

Proposition 8.4.2. The Samelson products of the canonical generators µi ∈π1(Ω(CP∞)K) satisfy the identities

[µi, µi]s = 0, [µi, µj ]s = 0 if and only if i, j ∈ K.Proof. By adjunction, we can work with the Whitehead products instead.

The Whitehead square [µi, µi]w is zero in π3((CP∞)K), because it is zero inπ3(CP∞) = 0. Furthermore, the map µi ∨ µj : S2 ∨ S2 → (CP∞)K with i 6= jextends to a map S2 × S2 → (CP∞)K whenever i, j is an edge of K, whichimplies that [µi, µj ]w = 0 whenever i, j ∈ K.


Corollary 8.4.3. The algebra H∗(Ω(CP∞)K;k) contains the subalgebra

(8.17) T 〈u1, . . . , um〉/(u2i = 0, uiuj + ujui = 0 if i, j ∈ K),where ui ∈ H1(Ω(CP∞)K;k) is the Hurewitz image of µi ∈ π1(Ω(CP∞)K).

The subalgebra above maps onto the ‘fully commutative’ algebra Λ[u1, . . . , um]under the projection map of (8.16).

In the homotopy fibration (8.15), since πk(Tm) = 0 for k > 1, any iterated

Samelson product of the form [µi1 , [µi2 , · · · [µik−1, µik ] · · · ]] with k > 1 composes

trivially into Tm and so lifts to ΩZK.The Whitehead product [µi, µj ]w : S3 → (CP∞)K is nontrivial whenever i, j

is a missing edge of K. We may generalise this construction by considering missingfaces I = i1, . . . , ik of K (recall that this means that I /∈ K, but any proper subsetof I is in K). Geometrically a missing face defines a subcomplex ∂∆(I) ⊂ K. Definethe k-fold higher Whitehead product [µi1 , . . . , µik ]w as the composite

(8.18) [µi1 , . . . , µik ]w : S2k−1 w−→ (S2)∂∆(I) −→ (CP∞)∂∆(I) −→ (CP∞)K

where (S2)∂∆(I) is the fat wedge of k spheres, w is the attaching map of the 2k-cellin the product (S2)I , and the last two maps of the polyhedral products are inducedby the inclusions S2 → CP∞ and ∂∆(I)→ K. The k-fold higher Samelson product[µi1 , . . . , µik ]s is defined as the adjoint of [µi1 , . . . , µik ]w:

[µi1 , . . . , µik ]s : S2k−2 −→ Ω(CP∞)K.

Remark. As it is standard with higher operations, the higher product[µi1 , . . . , µik ] is defined only when all shorter higher products of the µi1 , . . . , µik(corresponding to proper subsets of I) are trivial. The general definition of higherWhitehead and Samelson products (see [323])requires treatment of the indeter-minacy, which we avoided in the case of the polyhedral product (CP∞)K by thecanonical choice of map (8.18).

As in the case of standard (2-fold) products, higher Whitehead and Samelsonproducts of the µi can be iterated and lifted toΩZK. We summarise this observationas follows:

Proposition 8.4.4. Any iterated higher Whitehead product ν : Sp → (CP∞)K

of the canonical maps µi : S2 → (CP∞)K lifts to a map Sp → ZK.

Similarly, any iterated higher Samelson product ν : Sp−1 → Ω(CP∞)K of theµi : S

1 → Ω(CP∞)K lifts to a map Sp−1 → ΩZK.

Lifts Sp → ZK of higher iterated Whitehead products of the µi provide impor-tant family of spherical classes in H∗(ZK). We may ask the following question:

Problem 8.4.5. Assume that ZK is homotopy equivalent to a wedge of spheres.Is it true that all wedge summands are represented by lifts Sp → ZK of higheriterated Whitehead products of the canonical maps µi : S

2 → (CP∞)K?

For all known classes of examples when ZK is a wedge of spheres, the answerto the above question is positive. We shall give some evidence below.

Example 8.4.6.1. Let K be two points. The fibration (8.15) becomes

ΩS3 → Ω(CP∞ ∨ CP∞)→ S1 × S1,


and the corresponding sequence of Pontryagin algebras (8.16) is

0 −→ k[w]i−→ T 〈u1, u2〉/(u21, u22)

j−→ Λ[u1, u2] −→ 0

where k[w] = H∗(ΩS3;k), degw = 2, the map i takes w to the commutator

u1u2 + u2u1, and j is the projection to the quotient by the ideal generated byu1u2 + u2u1 (an exercise). So

H∗(Ω(CP∞ ∨ CP∞);k) = T 〈u1, u2〉/(u21, u22) = Λ[u1] ⋆ Λ[u2]

is the free product of two exterior algebras and i(k[w]) is its commutator subalgebra.In particular, exact sequence (8.16) does not split multiplicatively in this example.

Here u1, u2 are the Hurewitz images of µ1, µ2, the commutator u1u2 + u2u1 isthe Hurewitz image of the Samelson product [µ1, µ2]s : S

2 → Ω(CP∞∨CP∞), andw ∈ H2(ΩS

3) is the Hurewitz image of the lift of [µ1, µ2]s to ΩS3.2. Now let K = ∂∆2, the boundary of a triangle. The fibration (8.15) becomes

ΩS5 → Ω(CP∞)∂∆2 → T 3,

where Ω(CP∞)∂∆2

is the fat wedge of 3 copies of CP∞. We have H∗(ΩS5;k) =

k[w], degw = 4. Algebra (8.17) is isomorphic to Λ[u1, u2, u3], so the sequence ofPontryagin algebras (8.16) splits multiplicatively in this example:

0 −→ k[w] −→ k[w]⊗ Λ[u1, u2, u3] −→ Λ[u1, u2, u3] −→ 0.

Here w ∈ H4(Ω(CP∞)∂∆2

;k) is the Hurewicz image of the higher Samelson product

[µ1, µ2, µ3]s ∈ π4(Ω(CP∞)∂∆2

), which lifts to ΩS5. The fact that [µ1, µ2, µ3]s isa nontrivial higher Samelson product (and its Hurewicz image w is the ‘highercommutator product’ of u1, u2, u3) constitutes the additional information necessary

to distinguish between the topological monoids Ω(CP∞)∂∆2

and ΩS5 × T 3.This calculation generalises easily to the case K = ∂∆m−1, showing that

H∗(Ω(CP∞)K;k) ∼= k[w]⊗ Λ[u1, . . . , um]

where degw = 2m− 2.

Topological models for loop spaces. We consider the diagram

(8.19) DK(S1) : cat(K)→ tmon

whose value on the morphism I ⊂ J is the monomorphism of tori T I → T J . Theclassifying space diagram BDK(S

1) is DK(CP∞, pt) : cat(K) → top with colimit(CP∞)K. We denote the colimit and homotopy colimit of DK(S

1) by colimtmon

I∈K T I

and hocolimtmon

I∈K T I , respectively.

Theorem 8.4.7 ([261]). There is a commutative diagram

(8.20)

Ω hocolimtop

I∈KBTI g−−−−→

≃hocolimtmon

I∈K T I

Ωptop

y≃

yptmon

Ω(CP∞)K Ω colimtop

I∈KBTI −−−−→ colimtmon

I∈K T I

in Ho(tmon), where the top and left homomorphisms are homotopy equivalences.

Proof. We apply Corollary C.3.8 with D = DK(S1). The left projec-

tion Ωptop : Ω hocolimtop

I∈KBTI → Ω colimtop

I∈KBTI is a weak equivalence because

BDK(S1) = DK(CP∞, pt) is a cofibrant diagram in top.


Corollary 8.4.8. There is a weak equivalence

Ω(CP∞)K ≃ hocolimtmon

I∈K T I

in tmon.

The right projection ptmon (and therefore the bottom homomorphism in (8.20))is not a weak equivalence in general, because DK(S

1) is not a cofibrant diagramin tmon. The appropriate examples are discussed below.

Example 8.4.9.1. Let K be two points. Then

(CP∞)K = CP∞ ∨ CP∞, colimtmonDK(S1) = S1 ⋆ S1

where ⋆ denotes the free product of topological monoids, i.e. the coproductin tmon. The bottom homomorphism in (8.20) is Ω(CP∞ ∨ CP∞) → S1 ⋆ S1,it is a weak equivalence in tmon.

2. Now let K = ∂∆2. The loop space Ω(CP∞)∂∆2

is described in Exam-ple 8.4.6.2. On the other hand, the colimit of DK(S

1) is obtained by taking quotientof T 1 ⋆ T 2 ⋆ T 3 by the commutativity relations

t1 ⋆ t2 = t2 ⋆ t1, t2 ⋆ t3 = t3 ⋆ t2, t3 ⋆ t1 = t1 ⋆ t3

for ti ∈ T i, so that colimtmon

I∈K T I = T 3. It follows that the bottom map

Ω(CP∞)∂∆2 → T 3 in (8.20) is not a weak equivalence; it has kernel ΩS5.

The diagram D = DK(S1) : cat(K) → tmon is not cofibrant in this example.

Indeed, if we take I = 1, 2, then the induced diagram over the overcategorycat(K)↓I = cat(∆(I)) has the form

pt −−−−→ T 1

yy

T 2 −−−−→ T 1 × T 2

The map colimD|cat(∂∆(I)) → D(I) is the projection T 1 ⋆ T 2 → T 1 × T 2

from the free product to the cartesian product, which is not a cofibration in tmon.

Algebraic models for loop spaces. Our next aim is to obtain an algebraicanalogue of Theorem 8.4.7. We work over over a commutative ring k.

We define the face coalgebra k〈K〉 as the graded dual of the face ring k[K]. Asa k-module, k〈K〉 is free on generators vσ corresponding to multisets of m elementsof the form

σ = 1, . . . , 1︸︷︷︸k1

, 2, . . . , 2︸︷︷︸k2

, . . . ,m, . . . ,m︸︷︷︸km

such that support of σ (i.e. the set Iσ = i ∈ [m] : ki 6= 0) is a simplex of K. The

element vσ is dual to the monomial vk11 vk22 · · · vkmm ∈ k[K]. The comultiplicationtakes the form

∆vσ =∑

σ=τ ⊔ τ ′

vτ ⊗ vτ ′ ,

where the sum ranges over all partitions of σ into submultisets τ and τ ′.We recall the Adams cobar construction Ω∗ : dgc → dga, see (C.9), and the

Quillen functor L∗ : dgc → dgl, see (C.11). The loop algebra Ω∗k〈K〉 is our firstalgebraic model for Ω(CP∞)K:


Proposition 8.4.10. There is an isomorphism of graded algebras

H∗(Ω(CP∞)K;k) ∼= H(Ω∗k〈K〉) = Cotork〈K〉(k,k).

Proof. By dualising the integral formality results of [243, Theorem 4.8], weobtain a zigzag of quasi-isomorphisms

(8.21) C∗((CP∞)K;k)

≃←− · · · ≃−→ k〈K〉.in dgc (when k = Q this follows from Theorem 8.1.2). Since Ω∗ preserves quasi-isomorphisms, the zigzag above combines with Adams’ result (Theorem C.2.1) toobtain the required isomorphism of algebras.

Remark. When k is a field, there are isomorphisms

H∗(Ω(CP∞)K;k) ∼= Cotork〈K〉(k,k) ∼= Extk[K](k,k).

The graded algebra underlying the cobar construction Ω∗k〈K〉 is the tensor

algebra T (s−1k〈K〉) on the desuspended k-module k〈K〉= Ker(ε : k〈K〉 → k); the

differential is defined on generators by

d(s−1vσ) =∑

σ=τ⊔τ ′; τ, τ ′ 6=∅

s−1vτ ⊗ s−1vτ ′ ,

because d = 0 on k〈K〉. For future purposes it is convenient to write s−1vσ as χσfor any multiset σ.

We define now some algebraic diagrams over cat(K). Our previous algebraicdiagrams such as (8.1) were commutative, contravariant and cohomological, but toinvestigate the loop space Ω(CP∞)K we introduce models that are covariant andhomological. We consider the diagram

k[ · ]K : cat(K)→ dga, I 7→ k[vi : i ∈ I]which maps a morphism I ⊂ J to the monomorphism of polynomial algebrask[vi : i ∈ I] → k[vi : i ∈ J ] with deg vi = 2 and zero differential. Similarly, wedefine the diagrams

(8.22)

Λ[ · ]K : cat(K)→ dga, I 7→ Λ[ui : i ∈ I], deg ui = 1,

k〈 · 〉K : cat(K)→ dgc, I 7→ k〈vi : i ∈ I〉, deg vi = 2,

CL( · )K : cat(K)→ dgl, I 7→ CL(ui : i ∈ I), deg ui = 1,

where k〈vi : i ∈ I〉 denotes the free commutative coalgebra and CL(ui : i ∈ I)denotes the commutative Lie algebra on |I| generators.

The individual algebras and coalgebras in these diagrams are all commutative,but the context demands they be interpreted in the non-commutative categories;this is especially important when forming limits and colimits.

Note that colimdgc

k〈 ·〉K = k〈K〉, while colimdga Λ[ ·]K is the non-commutativealgebra (8.17) (an exercise).

Proposition 8.4.11. There are acyclic fibrations

Ω∗k〈vi : i ∈ I〉 ≃−→ Λ[ui : i ∈ I] and L∗k〈vi : i ∈ I〉 ≃−→ CL(ui : i ∈ I)in dga and dgl respectively, for any set I ⊂ [m].


Proof. We define the first map by χi 7→ ui for 1 6 i 6 m. Because

(8.23) dχii = χi ⊗ χi, dχij = χi ⊗ χj + χj ⊗ χi for i 6= j

hold in Ω∗k〈vi : i ∈ I〉, the map is consistent with the exterior relations in itstarget. So it is an epimorphism and quasi-isomorphism in dga, and hence an acyclicfibration. The corresponding result for dgl follows by restriction to primitives.

Observe that the diagram Λ[ · ]K in dga can be thought of as the diagram ofhomology algebras of topological monoids in the diagram DK(S

1), see (8.19), andthe diagram k〈 · 〉K in dgc is the diagram of homology coalgebras of spaces in theclassifying diagram BDK(S

1). This relationship extends to the following algebraicanalogue of Theorem 8.4.7:

Theorem 8.4.12 ([260]). There is a commutative diagram

(8.24)

Ω∗ hocolimdgc

k〈 · 〉K h−−−−→≃

hocolimdga Λ[ · ]K

Ω∗pdgc

y≃

ypdga

Ω∗k〈K〉 −−−−→ colimdga Λ[ · ]Kin Ho(dga), where the top and left arrows are isomorphisms.

Proof. This follows by considering the diagram

Ω∗ hocolimdgc

k〈 · 〉K hocolimdgaΩ∗k〈 · 〉K ≃−−−−→ hocolimdga Λ[ · ]KΩ∗p

dgc

y≃ pdga

y≃ pdga

y

Ω∗ colimdgc

k〈 · 〉K∼=←−−−− colimdgaΩ∗k〈 · 〉K −−−−→ colimdga Λ[ · ]K

‖Ω∗k〈K〉

Here the top right horizontal map is induced by the map of diagrams Ω∗k〈 · 〉K →Λ[ · ]K whose objectwise maps are acyclic fibrations from Proposition 8.4.11; themap of homotopy colimits is a weak equivalence because the map of diagrams isan acyclic Reedy fibration. The right square is commutative. The central verticalmap is a weak equivalence because the diagram Ω∗k〈 · 〉K is Reedy cofibrant. Thebottom left horizontal map is an isomorphism because Ω∗ is left adjoint. The leftvertical map is a weak equivalence because the diagram k〈 · 〉K is Reedy cofibrantand Ω∗ preserves weak equivalences. The resulting zigzag of quasi-isomorphisms

Ω∗ hocolimdgc

k〈 · 〉K ≃−→ · · · ≃−→ hocolimdga Λ[ · ]Kinduces an isomorphism in the homotopy category Ho(dga); we denote it by h.

As in the case of diagram (8.20), the right and bottom maps in (8.24) are notweak equivalences in general, because Λ[ · ]K is not a cofibrant diagram in dga.

The following statement is proved similarly.

Theorem 8.4.13 ([260]). There is a homotopy commutative diagram

(8.25)

L∗ hocolimdgc Q〈 · 〉K −−−−→

≃hocolimdgl CL( · )K

L∗pdgc

y≃

ypdgl

L∗Q〈K〉 −−−−→ colimdgl CL( · )K


in Ho(dgl), where the top and left arrows are isomorphisms.

The homotopy colimit decomposition above defines our second algebraic modelfor Ω(CP∞)K:

Corollary 8.4.14. For any simplicial complex K and commutative ring k,there are isomorphisms

H∗

(Ω(CP∞)K;k

) ∼= H(hocolimdga Λ[ · ]K

)

π∗(Ω(CP∞)K)⊗Z Q ∼= H

(hocolimdgl CL( · )K

)

of graded algebras and Lie algebras respectively.

Example 8.4.15.1. Let K be a discrete complex on m vertices, so that (CP∞)K is a wedge

of m copies of CP∞. The cobar construction Ω∗k〈K〉 on the corresponding facecoalgebra is generated as an algebra by the elements of the form χi...i with i ∈ [m].The first identity of (8.23) still holds, but χi⊗χj+χj⊗χi is no longer a boundaryfor i 6= j since there is no element χij in Ω∗k〈K〉. We obtain a quasi-isomorphism

Ω∗k〈K〉 ≃−→ Tk(u1, . . . , um)/(u2i = 0, 1 6 i 6 m)

that maps χi to ui. The right hand side is isomorphic to H∗(Ω(CP∞)K;k); it isthe coproduct (the free product) of m algebras Λ[ui]. Therefore, the bottom mapΩ∗k〈K〉 → colimdga Λ[ · ]K in (8.24) is a quasi-isomorphism in this example.

The algebra H∗(ΩZK;k) is the commutator subalgebra of H∗(Ω(CP∞)K;k)according to (8.16). In contains iterated commutators [ui1 , [ui2 , · · · [uik−1

, uik ] · · · ]]with k > 2 corresponding to iterated Samelson products in π∗(Ω(CP∞)K). On theother hand, ZK is a wedge of spheres given by (8.11). Therefore, H∗(ΩZK;k) is afree (tensor) algebra on the generators corresponding to the wedge summands. Sothe number of independent iterated commutators of length k is (k − 1)

(mk

). This

fact can be proved purely algebraically, see Corollary 8.5.8 below. There are nohigher Samelson products (and higher iterated commutators) in this example, as Kdoes not have missing faces with > 2 vertices.

2. Let K = ∂∆2. As we have seen in Example 8.4.6.2,

H∗(Ω(CP∞)K;k) ∼= k[w] ⊗ Λ[u1, u2, u3].

This can also be seen algebraically using the cobar model Ω∗k〈K〉. Here ui is thehomology classes of the element χi, and w ∈ H4(Ω(CP∞)K;k) is the homologyclass of the 4-dimensional cycle

ψ = χ1χ23 + χ2χ13 + χ3χ12 + χ12χ3 + χ13χ2 + χ23χ1,

whose failure to bound is due to the non-existence of χ123. Relations (8.23) hold,and give rise to the exterior relations between u1, u2, u3. Furthermore, a directcheck shows that χiψ−ψχi is a boundary, which implies that ui commutes with win H∗(Ω(CP∞)K;k) for 1 6 i 6 3.

Here the colimit of Λ[ ·]K in dga is obtained by taking quotient of Tk(u1, u2, u3)by all exterior relations. Therefore, colimdga Λ[ · ]K = Λ[u1, u2, u3] and the bottommapΩ∗k〈K〉 → colimdga Λ[ ·]K in (8.24) is not a quasi-isomorphism in this example.

3. Let K = sk1∂∆3, the 1-skeleton of a 3-simplex, or a complete graph on4 vertices. Arguments similar to those of the previous example show that the


Pontryagin algebra H∗(Ω(CP∞)K;k) contains 1-dimensional classes u1, . . . , u4 and4-dimensional classes w123, w124, w134, w234, corresponding to the four missingfaces with three vertices each. For example, w123 is the homology class of the cycleψ123 which may be thought of as the ‘boundary of the non-existing element χ123’.Identities (8.23) give rise to the exterior relations between u1, . . . , u4. We mayeasily check that ui commutes with wjkl if i ∈ j, k, l. There are four remainingnon-trivial commutators of the form [ui, wjkl] with all i, j, k, l different.

It follows that the commutator subalgebra H∗(ΩZK;k) contains four highercommutators wjkl = [uj , uk, ul] (the Hurewitz images of the higher Samelson prod-ucts [µj , µk, µl]s) and four iterated commutators [ui, wjkl]. On the other hand,Theorem 4.7.7 gives a homotopy equivalence

ZK ≃ (S5)∨4 ∨ (S6)∨3,

which implies that H∗(ΩZK;k) is a free algebra on four 4-dimensional and three5-dimensional generators. The point is that the commutators [ui, wjkl] are subjectto one extra relation, which can be derived as follows. Consider the relation

(8.26) dχ1234 = (χ1χ234 + χ234χ1) + · · ·+ (χ4χ123 + χ123χ4) + β

in Ω∗k〈v1, v2, v3, v4〉, where β consists of terms χσχτ such that |σ| = |τ | = 2.Denote the first four summands on the right hand side of (8.26) by α1, α2, α3,α4 respectively, and apply the differential to both sides. Observing that dα1 =−χ1ψ234 + ψ234χ1 = −[χ1, ψ234], and similarly for dα2, dα3 and dα4, we obtain

[χ1, ψ234] + [χ2, ψ134] + [χ3, ψ124] + [χ4, ψ123] = dβ.

The outcome is an isomorphism

H∗(Ω(CP∞)K;k) ∼= Tk(u1, u2, u3, u4, w123, w124, w134, w123)/I,where degwijk = 4 and I is generated by three types of relation:

· exterior algebra relations for u1, u2, u3, u4;· [ui, wjkl ] = 0 for i ∈ j, k, l;· [u1, w234] + [u2, w134] + [u3, w124] + [u4, w123] = 0.

As wijk is the higher commutator of ui, uj and uk, the third relation may beconsidered as a higher analogue of the Jacobi identity.

It is a challenging task to construct explicit algebraic models for Ω(CP∞)K

and ΩZK which would include a description of the Pontryagin algebra structure, aswell as higher Samelson and commutator products. The situation is considerablysimpler when K is a flag complex, as there are no higher products; this is the subjectof the next section.

Exercises.

8.4.16. For any 2n-dimensional (quasi)toric manifold M , show that there is afibration

ΩM −→ Ω(CP∞)K −→ T n,

which splits in top.

8.4.17. For the classes of simplicial complexes K described in Propositions 8.2.5and 8.2.6 (discrete complexes and trees), show that each wedge summand of ZK

is represented by a lift Sp → ZK of iterated Whitehead products of the µi : S2 →

(CP∞)K (no higher products appear here). Describe the corresponding iterated

8.5. THE CASE OF FLAG COMPLEXES 347

brackets explicitly. In particular, the answer to Problem 8.4.5 is positive for thesetwo classes of K.

8.4.18. Show that H∗(Ω(CP∞ ∨ CP∞);k) = T 〈u1, u2〉/(u21, u22) and de-scribe the sequence of Pontryagin algebras corresponding to the fibration ΩS3 →Ω(CP∞ ∨ CP∞)→ S1 × S1.

8.4.19. Show that colimdga Λ[ · ]K is the algebra given by (8.17).

8.5. The case of flag complexes

In this section we study the loop spaces associated with flag complexes K. Suchcomplexes have significantly simpler combinatorial properties, which are reflected inthe homotopy theory of the toric spaces. We modify results of the previous sectionin this context, and focus on applications to the Pontryagin rings and homotopyLie algebras of Ω(CP∞)K and ΩZK. We also describe completely the class of flagcomplexes K for which ZK is homotopy equivalent to a wedge of spheres.

For any simplicial complex K on [m], recall that a subset I ⊂ [m] is calleda missing face when every proper subset lies in K, but I itself does not. If everymissing face ofK has 2 vertices, thenK is a flag complex; equivalently, K is flag whenevery set of vertices that is pairwise connected spans a simplex. A flag complex istherefore determined by its 1-skeleton, which is a graph. When K is flag, we mayexpress the face ring as

k[K] = Tk(v1, . . . , vm)/(vivj − vjvi = 0 for i, j ∈ K, vivj = 0 for i, j /∈ K).

It is therefore quadratic, in the sense that it is the quotient of a free algebra byquadratic relations.

The following result of Froberg allows us to calculate the Yoneda algebrasExtA(k,k) explicitly for a class of quadratic algebras A that includes face rings offlag complexes.

Proposition 8.5.1 ([119, §3]). When k is a field and K is a flag complex,there is an isomorphism of graded algebras

(8.27) Extk[K](k,k) ∼= Tk(u1, . . . , um)/(u2i = 0, uiuj + ujui = 0 for i, j ∈ K).

Remark. The algebra on the right hand side of (8.27) is the quadratic dualof k[K]. A quadratic algebra A is called Koszul if its quadratic dual coincides withExtA(k,k), so Proposition 8.5.1 asserts that k[K] is Koszul when K is flag.

When K is flag, (8.17) is the whole Pontryagin algebra H∗(Ω(CP∞)K;k):

Theorem 8.5.2 ([260]). For any flag complex K, there are isomorphisms

H∗(Ω(CP∞)K;k) ∼= Tk(u1, . . . , um)/(u2i = 0, uiuj + ujui = 0 for i, j ∈ K)

π∗(Ω(CP∞)K)⊗Z Q ∼= FL(u1, . . . , um)/(

[ui, ui] = 0, [ui, uj] = 0 for i, j ∈ K),

where k is Z or a field, FL( · ) denotes a free Lie algebra and deg ui = 1.

Proof. By Proposition 8.4.10, H∗(Ω(CP∞)K;k) ∼= Cotork〈K〉(k,k). When k

is a field, Cotork〈K〉(k,k) ∼= Extk[K](k,k) by (C.10), and the first required isomor-phism follows from Proposition 8.5.1.

Now let k = Z. Denote by A algebra (8.17) with k = Z. Then A includes as asubalgebra in H∗(Ω(CP∞)K;Z) by Corollary 8.4.3. Consider the exact sequence

0→ Ai−→ H∗(Ω(CP∞)K;Z) −→ R→ 0


where R is the cokernel of i. By tensoring with a field k we obtain an exact sequence

A⊗ ki⊗k−→ H∗(Ω(CP∞)K;Z)⊗ k −→ R⊗ k→ 0

The composite map

A⊗ ki⊗k−→ H∗(Ω(CP∞)K;Z)⊗ k

j−→ H∗(Ω(CP∞)K;k)

is an isomorphism by the argument in the previous paragraph, and j is a monomor-phism by the universal coefficient theorem. Therefore, i⊗k is also an isomorphism,which implies that R⊗k = 0 for any field k. Thus, R = 0 and i is an isomorphism.

The isomorphism of Lie algebras follows by restriction to primitives.

The authors are grateful to Kouyemon Iriye and Jie Wu for pointing out that theoriginal argument of [260] for the theorem above can be extended to the case k = Z.

The associative algebra and the Lie algebra from Theorem 8.5.2 are exam-ples of graph products, by which one usually means algebraic objects described bygenerators corresponding to the vertices in a simple graph, with each edge givingrise to a commutativity relation between the generators corresponding to its twoends. These two graph products can be also described as the colimits of the dia-grams Λ[ · ]K and CL( · )K in dga and dgl respectively, see (8.22). It follows thatthe bottom maps in (8.24) and (8.25) are quasi-isomorphisms when K is flag, andhomotopy colimit in the models of Corollary 8.4.14 can be replaced by colimit:

Corollary 8.5.3. For any flag complex K, there are isomorphisms

H∗

(Ω(CP∞)K;k

) ∼= colimdga Λ[ · ]Kπ∗(Ω(CP∞)K

)⊗Z Q ∼= colimdgl CL( · )K

of graded algebras and Lie algebras respectively, where k = Z or a field.

Remark. The bottom homomorphism

Ω(CP∞)K → colimtmon

I∈K T I

in the topological model (8.20) is also a homotopy equivalence when K is flag,by [261, Proposition 6.3]. Furthermore, there are analogues of this result for otherpolyhedral powers. Interesting cases are (RP∞)K and (S1)K, for which there arehomotopy equivalence homomorphisms

(8.28)Ω(RP∞)K

≃−→ colimtmon

I∈K ZI2,

Ω(S1)K≃−→ colimtmon

I∈K ZI .

Since Z2 and Z are discrete groups, the colimit in tmon is the ordinary colimit ofgroups, and the two colimits above have the following graph product presentations:

colimgrp

I∈K ZI2 = F (g1, . . . , gm)/(g2i = 0, gigj = gjgi for i, j ∈ K),

colimgrp

I∈K ZI = F (g1, . . . , gm)/(gigj = gjgi for i, j ∈ K)

where F (g1, . . . , gm) denotes a free group on m generators. The two groups aboveare known as the right-angled Coxeter group and the right-angled Artin group cor-responding to the 1-skeleton (graph) of the flag complex K.

Homotopy equivalences (8.28) imply that the polyhedral power (RP∞)K is theclassifying space for the right-angled Coxeter group colimgrp

I∈K ZI2 and (S1)K is theclassifying space for the right-angled Artin group colimgrp

I∈K ZI . The former result is


implicit in the work of Davis and Januszkiewicz [90], and the latter is due to Kimand Roush [184]. Note that (S1)K is a finite cell complex.

Proposition 8.5.4. For a flag complex K, the Poincare series of the Pontrya-gin algebras H∗(Ω(CP∞)K;k) and H∗(ΩZK;k) are given by

F(H∗(Ω(CP∞)K;k);λ

)=

(1 + λ)n

1− h1λ+ · · ·+ (−1)nhnλn,

F(H∗(ΩZK;k);λ

)=

1

(1 + λ)m−n(1 − h1λ+ · · ·+ (−1)nhnλn),

where (h0, h1, . . . , hn) is the h-vector of K.

Proof. Since H∗(Ω(CP∞)K;k) is the quadratic dual of k[K], the identity

F(k[K];−λ

)· F(H∗(Ω(CP∞)K);λ

)= 1

follows from Froberg [119, §4] (in the identity above it is assumed that the gen-erators of k[K] have degree one). The Poincare series of the face ring is given byTheorem 3.1.10, whence the first formula follows. The second formula follows fromexact sequence (8.16).

The Poincare series of π∗(Ω(CP∞)K)⊗ZQ (and therefore the ranks of homotopygroups of (CP∞)K and ZK) can also be calculated in the flag case, although lessexplicitly. See [95, §4.2].

Example 8.5.5. Let K = ∂∆n, so that h0 = · · · = hn = 1. Then Ω(CP∞)K ≃ΩS2n+1 × T n+1, and

F(H∗(Ω(CP∞)K);λ

)=

(1 + λ)n+1

1− λ2n .

On the other hand, Proposition 8.5.4 gives

(1 + λ)n

1− λ+ λ2 + · · ·+ (−1)nλn =(1 + λ)n+1

1 + (−1)nλn+1.

The formulae agree if n = 1, in which case K is flag, but differ otherwise.

We recall from Section C.1 that a space X is coformal when its Quillen modelQ(X) is weakly equivalent to the rational homotopy Lie algebra π∗(ΩX) ⊗Z Q asobjects in dgl. The space (CP∞)K is always formal by Theorem 8.1.2, while itscoformality depends on K:

Theorem 8.5.6. The space (CP∞)K is coformal if and only if K is flag.

Proof. If K is flag, Theorem 8.4.12 together with Corollary 8.5.3 provide anacyclic fibration

Ω∗Q〈K〉 ∼= Ω∗ colimdgc Q〈 · 〉K ≃−→ colimdga Λ[ · ]K

in dgaQ. Restricting to primitives yields a quasi-isomorphism e : L∗Q〈K〉 ≃−→π∗(Ω(CP∞)K)⊗Z Q in dgl.

Now choose a minimal model MK → Q[K] for the face ring in cdgaQ. Itsgraded dual Q〈K〉 → CK is a minimal model for Q〈K〉 in cdgc (see [241, §5]), soΩ∗Q〈K〉 → Ω∗CK is a weak equivalence in dga. Restricting to primitives providesthe central map in the zigzag

(8.29) LK≃−→ L∗CK

≃←− L∗Q〈K〉 e−→ π∗(Ω(CP∞)K)⊗Z Q


of quasi-isomorphisms in dgl, where LK is a minimal model for (CP∞)K in dgl[241, §8]. Hence (CP∞)K is coformal.

On the other hand, every missing face of K with > 2 vertices determines anontrivial higher Samelson bracket in π∗(Ω(CP∞)K) ⊗Z Q. The existence of suchbrackets in π∗(ΩX)⊗ZQ ensures that X cannot be coformal, just as higher Masseyproducts in H∗(X ;Q) obstruct formality.

Unlike the situation with the Pontryagin algebraH∗(Ω(CP∞)K), we are unableto describe the structure of its commutator subalgebraH∗(ΩZK) completely even inthe flag case. However, the following result identifies a minimal set of multiplicativegenerators as a specific set of iterated commutators of the ui:

Theorem 8.5.7 ([131, Theorem 4.3]). Assume that K is flag and k is a field.The algebra H∗(ΩZK;k), viewed as the commutator subalgebra of (8.17) via ex-

act sequence (8.16), is multiplicatively generated by∑

I⊂[m] dim H0(KI) iterated

commutators of the form

[uj, ui], [uk1 , [uj , ui]], . . . , [uk1 , [uk2 , · · · [ukm−2 , [uj , ui]] · · · ]]where k1 < k2 < · · · < kp < j > i, ks 6= i for any s, and i is the smallestvertex in a connected component not containing j of the subcomplex Kk1,...,kp,j,i.Furthermore, this multiplicative generating set is minimal, that is, the commutatorsabove form a basis in the submodule of indecomposables in H∗(ΩZK;k).

Remark. To help clarify the statement of Theorem 8.5.7, it is useful to considerwhich brackets [uj, ui] are in the list of multiplicative generators for H∗(ΩZK;k).If j, i ∈ K then i and j are in the same connected component of the subcomplexKj,i, so [uj, ui] is not a multiplicative generator. On the other hand, if j, i /∈ Kthen the subcomplex Kj,i consists of the two distinct points i and j, and i is thesmallest vertex in its connected component of Kj,i which does not contain j, so[uj, ui] is a multiplicative generator.

For a given I = k1, . . . , kp, j, i, the number of the commutators containing all

uk1 , . . . , ukp , uj , ui in the set above is equal to dim H0(KI) (one less the number of

connected components in KI), so there are indeed∑I⊂[m] dim H0(KI) commutators

in total. More details are given in examples below.

An important particular case of Theorem 8.5.7 corresponds to K consisting ofm disjoint points. This result may be of independent algebraic interest, as it isan analogue of the description of a basis in the commutator subalgebra of a freealgebra, given by Cohen and Neisendorfer [80]:

Corollary 8.5.8. Let A be the commutator subalgebra of the algebraT 〈u1, . . . , um〉/(u2i = 0), that is, A is the algebra defined by the exact sequence

1 −→ A −→ T 〈u1, . . . , um〉/(u2i = 0) −→ Λ[u1, . . . , um] −→ 1

where deg ui = 1. Then A is a free associative algebra minimally generated by theiterated commutators of the form

[uj, ui], [uk1 , [uj , ui]], . . . , [uk1 , [uk2 , · · · [ukm−2 , [uj , ui]] · · · ]]where k1 < k2 < · · · < kp < j > i and ks 6= i for any s. Here, the number ofcommutators of length ℓ is equal to (ℓ − 1)

(mℓ

).


5

1

2

3

4

qqqqq

❩

❩❩❩❩

Figure 8.1. Boundary of pentagon.

Example 8.5.9. Let K be the boundary of pentagon, shown in Fig. 8.1.Theorem 8.5.7 gives the following 10 generators for the algebra H∗(ΩZK):

a1 = [u3, u1], a2 = [u4, u1], a3 = [u4, u2], a4 = [u5, u2], a5 = [u5, u3],

b1 = [u4, [u5, u2]], b2 = [u3, [u5, u2]], b3 = [u1, [u5, u3]],

b4 = [u3, [u4, u1]], b5 = [u2, [u4, u1]],

where deg ai = 2 and deg bi = 3. In the notation of the beginning of the previoussection, a1 is the Hurewicz image of the Samelson product [µ3, µ1] : S

2 → Ω(CP∞)K

lifted to ΩZK, and b1 is the Hurewicz image of the iterated Samelson product[µ4, [µ5, µ2]] : S

3 → Ω(CP∞)K lifted to ΩZK; the other ai and bi are describedsimilarly. We therefore have adjoint maps

ι : (S2 ∨ S3)∨5 → ΩZK and j : (S3 ∨ S4)∨5 → ZK

corresponding to the wedge of all ai and bi. Now a calculation using relations fromTheorem 8.5.2 and the Jacobi identity shows that ai and bi satisfy the relation

(8.30) − [a1, b1] + [a2, b2] + [a3, b3]− [a4, b4] + [a5, b5] = 0,

where [ai, bi] = aibi − biai. (One can make all the commutators to enter the sumwith positive signs by changing the order the elements in the commutators definingai, bi.) This relation has a topological meaning. In general, suppose that M and Nare d-dimensional manifolds. Let M be the (d− 1)-skeleton of M , or equivalently,M is obtained from M by removing a disc in the interior of the d-cell of M . DefineN similarly. Suppose that f : Sd−1 → M and g : Sd−1 → N are the attachingmaps for the top cells in M and N . Then the attaching map for the top cell in the

connected sum M #N is Sd−1 f+g−→M ∨N . In our case, S3 × S4 is a manifold andthe attaching map S6 → S3 ∨S4 for its top cell is the Whitehead product [s1, s2]w,where s1 and s2 respectively are the inclusions of S3 and S4 into S3 ∨ S4. Theattaching map for the top cell of the 5-fold connected sum (S3×S4)#5 is thereforethe sum of five such Whitehead products. Composing it with j into ZK and passingto the adjoint map we obtain

∑5i=1±[ai, bi] (the signs depend on the orientation

chosen, see Construction D.3.9). By (8.30), this sum is null homotopic. Thus theinclusion j : (S3 ∨ S4)∨5 → ZK extends to a map

j : (S3 × S4)#5 → ZK.

Furthermore, a calculation using Theorem 4.5.4 shows that j induces an isomor-phism in cohomology (see Example 4.6.10), that is, j is a homotopy equivalence.Since both (S3 × S4)#5 and ZK are manifolds, the complement of (S3 ∨ S4)∨5 in


(S3 × S4)#5 and ZK is a 7-disc, so that the extension map j can be chosen to be

one-to-one, which implies that j is a homeomorphism.We also obtain that H∗(ΩZK) is the quotient of a free algebra on ten generators

ai, bi by relation (8.30). Its Poicare series is given by Proposition 8.5.4:

P(H∗(ΩZK);λ

)=

1

1− 5λ2 − 5λ3 + λ5.

The summand t5 in the denominator is what differs the Poincare series of the one-relator algebra H∗(ΩZK) from that of the free algebra H∗(Ω(S3 ∨ S4)∨5).

A similar argument can be used to show that ZK is homeomorphic to a con-nected sum of sphere products when K is a boundary of a m-gon with m > 4,therefore giving a homotopical proof of a particular case of Theorem 4.6.12. Itwould be interesting to give a homotopical proof of this theorem in general.

Another result of [131] identifies the class of flag complexes K for which ZK

has homotopy type of a wedge of spheres.We recall from Definition 4.9.5 that k[K] is a Golod ring and K is a

Golod complex when the multiplication and all higher Massey products inTork[v1,...,vm](k[K],k) = H(Λ[u1, . . . , um]⊗ k[K], d) are trivial.

We also need some terminology from graph theory. Let Γ be a graph on thevertex set [m]. A clique of Γ is a subset I of vertices such that every two verticesin I are connected by an edge. Each flag complex K is the clique complex of itsone-skeleton Γ = K1, that is, the simplicial complex formed by filling in each cliqueof Γ by a face.

A graph Γ is called chordal if each of its cycles with > 4 vertices has a chord(an edge joining two vertices that are not adjacent in the cycle). Equivalently, achordal graph is a graph with no induced cycles of length more than three.

The following result gives an alternative characterisation of chordal graphs.

Theorem 8.5.10 (Fulkerson–Gross [121]). A graph is chordal if and only if itsvertices can be ordered in such a way that, for each vertex i, the lesser neighboursof i form a clique.

Such an order of vertices is called a perfect elimination order .

Theorem 8.5.11 ([131]). Let K be a flag complex and k a field. The followingconditions are equivalent:

(a) k[K] is a Golod ring;(b) the multiplication in H∗(ZK;k) is trivial;(c) Γ = K1 is a chordal graph;(d) ZK has homotopy type of a wedge of spheres.

Proof. (a)⇒(b) This is by definition of Golodness and Theorem 4.5.4.(b)⇒(c) Assume that K1 is not chordal, and choose an induced chordless cycle I

with |I| > 4. Then the full subcomplex KI is the same cycle (the boundary of an |I|-gon), and therefore ZKI

is a connected sum of sphere products. Hence, H∗(ZKI)

has nontrivial products (this can be also seen directly by using Theorem 4.5.8).Then, by Theorem 4.5.8, the same nontrivial products appear in H∗(ZK).

(c)⇒(d) Assume that the vertices of K are in perfect elimination order. Weassign to each vertex i the clique Ii consisting of i and the lesser neighbours of i.Each maximal face of K (that is, each maximal clique of K1) is obtained in this way,


so we get an induced order on the maximal faces: Ii1 , . . . , Iis . Then, for each k =1, . . . , s, the simplicial complex

⋃j<k Iij is flag (since it is the full subcomplex

K1,2,...,ik−1 in a flag complex). The intersection(⋃

j<k Iij)∩Iik is a clique, so it

is a face of⋃j<k Iij . Therefore, ZK has homotopy type of a wedge of spheres by

Corollary 8.2.4.(d)⇒(a) This is by definition of the Golod property and the fact that the

cohomology of the wedge of spheres contains only trivial Massey products.

Corollary 8.5.12. Assume that K is flag with m vertices, and ZK has homo-topy type of a wedge of spheres. Then

(a) the maximal dimension of spheres in the wedge is m+ 1;(b) the number of spheres of dimension ℓ + 1 in the wedge is given by∑

|I|=ℓ dim H0(KI), for 2 6 ℓ 6 m;

(c) Hi(KI) = 0 for i > 0 and all I.

Proof. If ZK is a wedge of spheres, then H∗(ΩZK) is a free algebra on gen-erators described by Theorem 8.5.7, which implies (a) and (b). It also follows that

H∗(ZK) ∼=⊕

J⊂[m] H0(KJ ). On the other hand, H∗(ZK) ∼=

⊕J⊂[m] H

∗(KJ ) by

Theorem 4.5.8, whence (c) follows.

Remark. The equivalence of (a), (b) and (c) in Theorem 8.5.11 was provedin [28]. All the implications in the proof of Theorem 8.5.11 except (c)⇒(d) are validfor arbitrary K, with the same arguments. However, (c)⇒(d) fails in the non-flagcase. Indeed, if K be the triangulation of RP 2 from Example 3.2.12.4, then K1 isa complete graph, so it is chordal. However, ZK is not homotopy equivalent to awedge of spheres, because it has 2-torsion in homology. Furthermore, this K is aGolod complex by Exercise 4.9.7. The following question is still open:

Problem 8.5.13 ([131]). Assume that H∗(ZK) has trivial multiplication, sothat K is Golod, over any field. Is it true that ZK is a co-H-space, or even asuspension, as in all known examples?

Exercises.

8.5.14. Shown that the generators a1, . . . , a5, b1, . . . , b5 of the Pontryagin alge-bra H∗(ΩZK) from Example 8.5.9 satisfy relation (8.30).

8.5.15 ([131, Example 3.3]). Describe explicitly the homotopy type of ZK whenK is the triangulation of RP 2 from Example 3.2.12.4.

CHAPTER 9

Torus actions and complex cobordism

Here we consider applications of toric methods in the theory of complex cobor-dism. In particular, we describe new families of toric generators of complex bordismring and quasitoric representatives in bordism classes. We also develop the theoryof torus-equivariant genera with applications to rigidity and fibre multiplicativityproblems, and provide explicit formulae for bordism classes and genera of quasitoricmanifolds and their generalisations via localisation techniques.

We refer to Appendices D and E for the background material on complex(co)bordism and Hirzebruch genera.

As usual, when working with cobordism we assume all manifolds to be smoothand compact. We denote by [M ] the bordism class in ΩU2n of 2n-dimensional stablycomplex manifold M , and denote by 〈M〉 ∈ H2n(M) its fundamental homologyclass defined by the orientation arising from the stably complex structure.

9.1. Toric and quasitoric representatives in complex bordism classes

Describing multiplicative generators for the complex bordism ring ΩU and rep-resenting bordism classes by manifolds with specific nice properties are well-knownquestions in cobordism theory. For the application of toric methods, it is importantto represent complex bordism classes by manifolds with nicely behaving torus ac-tions preserving the stably complex structure. In the context of oriented bordism,this question goes back to the fundamental work of Conner–Floyd [81].

The most ‘nicely behaving’ torus actions that we have at our disposal are thetorus actions on toric and quasitoric manifolds. Such an action does not exist onMilnor hypersurfacesHij , which constitute the most well-known multiplicative gen-erator set for ΩU (see Theorem 9.1.4). An alternative multiplicative generator setfor ΩU consisting of projective toric manifolds Bij was constructed by Buchstaberand Ray in [60]. Each Bij is a complex projectivisation of a sum of line bundlesover the bounded flag manifold Bi; in particular, Bij is a generalised Bott manifold.

It seems likely that a minimal set of ring generators of ΩU can be found amongtoric manifolds, i.e. there exist projective toric manifolds Xi whose bordism classesai = [Xi] are polynomial generators of the bordism ring: ΩU ∼= Z[a1, a2, . . .]. Apartial result in this direction was obtained by Wilfong [322]. Nevertheless, not anycomplex bordism class can be represented by a toric manifold. The reason is thattoric manifolds are very special algebraic varieties, and there are many restrictionson their characteristic numbers. For example, the Todd genus of a toric manifold isequal to 1, which implies that the bordism class of a disjoint union of toric manifoldscannot be represented by a toric manifold.

The main result of this section is Theorem 9.1.16 (originally proved in [58]),which shows that any complex bordism class (in dimensions > 2) contains a qua-sitoric manifold. A canonical torus-invariant stably complex structure is induced

355

356 9. TORUS ACTIONS AND COMPLEX COBORDISM

by an omniorientation of a quasitoric manifold, see Corollary 7.3.15. The abovementioned result of [60] provides an additive basis for each bordism group ΩU2nrepresented by toric manifolds; it implies that any complex bordism class can berepresented by a disjoint union of toric manifolds. The next step is to replace dis-joint unions by connected sums. There is the standard construction of connectedsum of T -manifolds at their fixed points. (The connected sum of two stably complexmanifolds M1 and M2 always admits a stably complex structure representing thebordism class [M1]+ [M2].) Davis–Januzkiewicz [90] proposed to use this construc-tion to make the connected sum M1 #M2 of two quasitoric manifolds M1 and M2

into a quasitoric manifold over the connected sum P1 # P2 of quotient polytopes.However, the main difficulty here is that one needs to keep track of both the torusaction and the stably complex structure on the connected sum of manifolds. Itturns out that the connected sum M1 #M2 does not always admit an omniorien-tation such that the bordism class [M1 #M2] ∈ ΩU of the induced T k-invariantstably complex structure represents the sum [M1] + [M2]; this depends on the signpattern of fixed points of the manifolds.

In order to overcome the difficulty describe above, we replace M2 by a bordantquasitoric manifold M ′

2 whose quotient polytope is In # P2 (a connected sum ofP2 with an n-cube). Then we show that the connected sum M1 #M ′

2 admits astably complex structure which is invariant under the torus action and representsthe bordism class [M1] + [M ′

2] = [M1] + [M2]; the quotient polytope of M1 #M ′2 is

P1 # In # P2. This allows us to finish the proof of the main result.This result on quasitoric representatives can be viewed as an answer to a toric

version of the famous Hirzebruch question (see Problem D.5.10) on bordism classesrepresentable by connected nonsingular algebraic varieties. Note that quasitoricmanifolds are connected by definition.

Using the constructions of Chapter 6 we can interpret this result as follows:each complex bordism class can be represented by the quotient of a nonsingularcomplete intersection of real quadrics by a free torus action.

Milnor hypersurfaces Hij are not quasitoric. Milnor hypersurfaces

Hij = (z0 : · · · : zi)× (w0 : · · · : wj) ∈ CP i × CP j : z0w0 + · · ·+ ziwi = 0(corresponding to pairs of integers j > i > 0) form the most well-known set ofmultiplicative generators of the complex bordism ring ΩU , see Theorem D.5.7.However, as it was shown in [60], the manifold Hij is not (quasi)toric when i > 1.We give the argument below.

Construction 9.1.1. Let Ci+1 ⊂ Cj+1 be the subspace generated by the firsti + 1 vectors of the standard basis of Cj+1. We identify CP i with the set of linesl ⊂ Ci+1. To each line l we assign the set of hyperplanes W ⊂ Cj+1 containing l.This set can be identified with CP j−1. Consider the set of pairs

E =(l,W ) : l ⊂W, l ⊂ Ci+1, W ⊂ Cj+1

.

The projection (l,W ) 7→ l defines a bundle E → CP i with fibre CP j−1.

Lemma 9.1.2. Milnor hypersurface Hij is identified with the space E above.

Proof. Indeed, a line l ⊂ Ci+1 can be given by its generating vector withhomogeneous coordinates (z0 : z1 : · · · : zi). A hyperplane W ⊂ Cj+1 is given by a

9.1. TORIC REPRESENTATIVES IN COMPLEX BORDISM CLASSES 357

linear form with coefficients w0, w1, . . . , wj . The condition l ⊂ W is equivalent tothe equation in the definition of Hij .

Theorem 9.1.3. The cohomology ring of Hij is given by

H∗(Hij) ∼= Z[u, v]/(ui+1, (ui + ui−1v + · · ·+ uvi−1 + vi)vj−i

),

where deg u = deg v = 2.

Proof. We use the notation from Construction 9.1.1. Let ζ denotes the vectorbundle over CP i whose fibre over l ∈ CP i is the j-plane l⊥ ⊂ Cj+1. Then Hij isidentified with the projectivisation CP (ζ). Indeed, for any line l′ ⊂ l⊥ representinga point in the fibre of the bundle CP (ζ) over l ∈ CP i, the hyperplane W = (l′)⊥ ⊂Cj+1 contains l, so that the pair (l,W ) defines a point in Hij by Lemma 9.1.2.The rest of the proof reproduces the general argument for the description of thecohomology of a complex projectivisation (see Theorem 7.8.2).

Denote by η the tautological line bundle over CP i (its fibre over l ∈ CP i is theline l). Then η ⊕ ζ is a trivial (j + 1)-plane bundle. Set w = c1(η) ∈ H2(CP i) andconsider the total Chern class c(η) = 1 + c1(η) + c2(η) + · · · . Since c(η)c(ζ) = 1and c(η) = 1− w, it follows that

(9.1) c(ζ) = 1 + w + · · ·+ wi.

Consider the projection p : CP (ζ) → CP i. Denote by γ the tautological linebundle over CP (ζ), whose fibre over l′ ∈ CP (ζ) is the line l′. Let γ⊥ denotethe (j − 1)-plane bundle over CP (ζ) whose fibre over l′ ⊂ l⊥ is the orthogonalcomplement to l′ in l⊥ (by definition, a point of CP (ζ) is represented by a linel′ in a fibre l⊥ of bundle ζ). It is easy to see that p∗(ζ) = γ ⊕ γ⊥. We setv = c1(γ) ∈ H2(CP (ζ)) and u = p∗(w) ∈ H2(CP (ζ)). Then ui+1 = 0. We havec(γ) = 1− v and c(p∗(ζ)) = c(γ)c(γ⊥), hence

c(γ⊥) = p∗(c(ζ)

)(1− v)−1 = (1 + u+ · · ·+ ui)(1 + v + v2 + · · · ),

see (9.1). Since γ⊥ is a (j−1)-plane bundle, it follows that cj(γ⊥) = 0. Calculating

the homogeneous component of degree j in the identity above, we obtain the second

relation vj−i∑ik=0 u

kvi−k = 0. It follows that there is a homomorphism Z[u, v]→H∗(CP (ζ)) which factors through a homomorphism R → H∗(CP (ζ)), where R isthe quotient ring of Z[u, v] given in the theorem.

It remains to observe that R → H∗(CP (ζ)) is actually an isomorphism. Thisfollows by considering the Serre spectral sequence of the bundle p : CP (ζ)→ CP i.Since both CP i and CP j−1 have only even-dimensional cells, the spectral sequencecollapses at E2. It follows that Z[u, v] → H∗(CP (ζ)) is an epimorphism, andtherefore so is R → H∗(CP (ζ)). Furthermore, the cohomology groups of CP (ζ)are the same as those of CP i × CP j−1, which implies that R → H∗(CP (ζ)) is amonomorphism.

Theorem 9.1.4. There is no torus action on the Milnor hypersurface Hij withi > 1 making it into a quasitoric manifold.

Proof. The cohomology of a quasitoric manifold has the form Z[v1, . . . , vm]/I,where I is the sum of two ideals, one generated by square-free monomials and an-other generated by linear forms (see Theorem 7.3.27). We may assume that thecharacteristic matrix Λ has reduced form (7.6) and express the first n generators


v1, . . . , vn via the last m − n ones by means of linear relations with integer coeffi-cients. Therefore, we have

Z[v1, . . . , vm]/I ∼= Z[w1, . . . , wm−n]/I ′,where I ′ is an ideal with basis consisting of products of > 2 integers linear forms.

Assume now that Hij is a quasitoric manifold. Then we have

Z[w1, . . . , wm−n]/I ′ ∼= Z[u, v]/I ′′,where I ′′ is the ideal from Theorem 9.1.3. Comparing the dimensions of linear(degree-two) components above, we obtain m − n = 2, so that w1, w2 can beidentified with u, v after a linear change of variables. Thus, the ideal I ′′ has a basisconsisting of polynomials which are decomposable into linear factors over Z, whichis impossible for i > 1.

Remark. Hij is a projectivisation of a complex j-plane bundle over CP i, butthis bundle does not split into a sum of line bundles, preventingHij from carrying aneffective T i+j−1-action. See the remark after Definition 7.8.26 and Exercise 9.1.23.

Toric generators for the bordism ring ΩU . Here we describe, follow-ing [60] and [58], a family of toric manifolds Bij , 0 6 i 6 j satisfying thecondition si+j−1[Bij ] = si+j−1[Hij ], where sn[M ] denotes the characteristic num-ber defined by (D.8). This implies that the family Bij multiplicatively generatesthe complex bordism ring, by the same argument as Theorem D.5.7.

Construction 9.1.5. Given a pair of integers 0 6 i 6 j, we introduce themanifold Bij consisting of pairs (U ,W ), where

U = U1 ⊂ U2 ⊂ · · · ⊂ Ui+1 = Ci+1, dimUk = kis a bounded flag in Ci+1 (that is, Uk ⊃ Ck−1, see Construction 7.7.1) and W isa hyperplane in Cj+1 containing U1. The projection (U ,W ) 7→ U describes Bij asthe projectivisation of a j-plane bundle over the bounded flag manifold BF i. Thisbundle splits into a sum of line bundles:

Bij = CP (ρ11 ⊕ · · · ⊕ ρ1i ⊕ Cj−i),

where ρ11, . . . , ρ1i are the line bundles over BF i described in Proposition 7.7.7 (the

splitting follows from the fact that W can be identified with a line in U⊥1 ⊕ Cj−i

using the Hermitian scalar product in Cj+1). Therefore, Bij is a generalised Bottmanifold (see Definition 7.8.26). It follows that Bij is a toric manifold, and alsoa quasitoric manifold over the combinatorial polytope Ii ×∆j−1. The descriptionof the corresponding characteristic matrix and characteristic submanifolds can befound in [62, Examples 2.9, 4.5] or [58, Example 3.13].

Proposition 9.1.6. Let f : BF i → CP i be the map sending a bounded flag Uto its first line U1 ⊂ Ci+1. Then the bundle Bij → BF i is induced from the bundleHij → CP i by means of the map f :

Bij −−−−→ Hijyy

BF if−−−−→ CP i

.

Proof. This follows from Lemma 9.1.2.


Theorem 9.1.7. We have si+j−1[Bij ] = si+j−1[Hij ], where the characteristicnumber si+j−1 of Milnor hypersurface Hij is given by Lemma D.5.6.

Proof. We first prove a lemma:

Lemma 9.1.8. Let f : M → N be a degree d map of 2i-dimensional stablycomplex manifolds, and let ξ be a complex j-plane bundle over N , j > 1. Then

si+j−1[CP (f∗ξ)] = d · si+j−1[CP (ξ)].

Proof. Let p : CP (ξ) → N be the projection, γ the tautological bundle overCP (ξ), and γ⊥ the complementary bundle, so that γ ⊕ γ⊥ = p∗(ξ). Then we have

T (CP (ξ)) = p∗TN ⊕ TF (CP (ξ)),where TF (CP (ξ)) is the tangent bundle along the fibres of the projection p. SinceTF (CP (ξ)) = Hom(γ, γ⊥) and Hom(γ, γ) = C, it follows that

TF (CP (ξ)) ⊕ C = Hom(γ, γ ⊕ γ⊥).Therefore,

(9.2) T (CP (ξ))⊕ C = p∗TN ⊕Hom(γ, γ ⊕ γ⊥) == p∗TN ⊕Hom(γ, p∗ξ) = p∗TN ⊕ (γ ⊗ p∗ξ),

where γ = Hom(γ,C).

The map f : M → N induces a map f : CP (f∗ξ)→ CP (ξ) such that

· pf = fp1, where p1 : CP (f∗ξ)→M is the projection;

· deg f = deg f ;

· f∗γ is the tautological bundle over CP (f∗ξ).

Using (9.2), we calculate

si+j−1

(T (CP (ξ))

)= p∗si+j−1(TN) + si+j−1(γ ⊗ p∗ξ) = si+j−1(γ ⊗ p∗ξ)

(since i+ j − 1 > i), and similarly for T (CP (f∗ξ)). Thus,

si+j−1[CP (f∗ξ)] = si+j−1

(T (CP (f∗ξ))

)⟨CP (f∗ξ)

⟩

= si+j−1

((f∗γ)⊗ p∗1f∗ξ

)⟨CP (f∗ξ)

⟩= si+j−1

(f∗(γ ⊗ p∗ξ)

)⟨CP (f∗ξ)

⟩

= si+j−1(γ ⊗ p∗ξ)f∗〈CP (f∗ξ)〉 = d · si+j−1(γ ⊗ p∗ξ)〈CP (ξ)〉= d · si+j−1[CP (ξ)].

To finish the proof of Theorem 9.1.7 we note that the map f : BF i → CP i

from Proposition 9.1.6 has degree 1. (The map f : BF i → CP i is birational: itis an isomorphism on the affine chart V 0,...,0 = U ∈ BF i : U1 6⊂ Ci, because abounded flag in V 0,...,0 ⊂ BF i is uniquely determined by its first line U1.)

Theorem 9.1.9 ([60]). The bordism classes of toric varieties Bij, 0 6 i 6 j,multiplicatively generate the complex bordism ring ΩU∗ . Therefore, every complexbordism class contains a disjoint union of toric manifolds.

Proof. The first statement follows from Theorems 9.1.7 and D.5.7. A productof toric manifolds is toric, but a disjoint union of toric manifolds is not, since toricmanifolds are connected by definition.


Remark. The manifolds Hij and Bij are not bordant in general, althoughH0j = B0j = CP j−1 and H1j = B1j . The proof of Lemma 9.1.8 uses specific prop-erties of the number sn, and it does not work for arbitrary characteristic numbers.

Connected sums. Our next goal is to replace the disjoint union of toric man-ifolds by a version of connected sum, which will be a quasitoric manifold.

Construction 9.1.10 (Equivariant connected sum at fixed points). We givethe construction for quasitoric manifolds only, although it can be generalised easilyto locally standard T -manifolds. Let M ′ = M(P ′, Λ′′) and M ′′ = M(P ′′, Λ′′) betwo quasitoric manifolds over n-polytopes P ′ and P ′′, respectively (see Section 7.3).We assume that both characteristic matrices Λ′ and Λ′′ are in the refined form (7.6),and denote by x′ and x′′ the initial vertices (given by the intersection of the first nfacets) of P ′ and P ′′, respectively.

Consider the connected sum of polytopes P ′#P ′′ = P ′#x′,x′′P ′′ (see Construc-tion 1.1.13). By definition, the equivariant connected sum M ′#M ′′ =M ′#x′,x′′M ′′

is the quasitoric manifold over P ′ # P ′′ with characteristic matrix

(9.3) Λ# =

1 0 · · · 0 λ′1,n+1 · · · λ′1,m′ λ′′1,n+1 · · · λ′′1,m′′

0 1 · · · 0 λ′2,n+1 · · · λ′2,m′ λ′′2,n+1 · · · λ′′2,m′′

......

. . ....

.... . .

......

. . ....

0 0 · · · 1 λ′n,n+1 · · · λ′n,m′ λ′′n,n+1 · · · λ′′n,m′′

.

Note that the matrix Λ# is not refined, because the first n facets of P ′ # P ′′ donot intersect.

The manifold M ′#M ′′ is T n-equivariantly diffeomorphic to the manifold ob-tained by removing from M ′ and M ′′ invariant neighbourhoods of the fixed pointscorresponding to x′ and x′′ with subsequent T n-equivariant identification of theboundaries of these neighbourhoods. The latter manifold becomes the standardconnected sum M ′ #M ′′ (see Construction D.3.9) if we forget the action.

In order to define an omniorientation (and therefore an invariant stably complexstructure) on M ′#M ′′ we need to specify an orientation of M ′#M ′′ along withmatrix (9.3).

Since both M ′ and M ′′ are oriented, the quasitoric manifold M ′#M ′′ can beoriented so as to be oriented diffeomorphic either to the oriented connected sumM ′ #M ′′, or to M ′ #M ′′ (see Construction D.3.9). In the first case we say thatthe orientation of M ′#M ′′ is compatible with the orientations of M ′ and M ′′.

The existence of a compatible orientation on M ′#M ′′ can be detected from thecombinatorial quasitoric pairs (P ′, Λ′) and (P ′′, Λ′′). We recall the notion of sign ofa fixed point of a quasitoric manifold M (or a vertex of the quotient polytope P ).By Lemma 7.3.18, the sign σ(x) of a vertex x = Fj1 ∩ · · · ∩ Fjn measures thedifference between the orientations of TxM and (ρj1 ⊕ · · · ⊕ ρjn)|x. Therefore,

σ(x) = vj1 · · · vjn〈M〉,where vi = c1(ρi) ∈ H2(M), 1 6 i 6 m, are the ring generators of H∗(M) and〈M〉 ∈ H2n(M) is the fundamental homology class.

Lemma 9.1.11. The equivariant connected sum M ′#x′,x′′M ′′ of omniorientedquasitoric manifolds admits an orientation compatible with the orientations of M ′

and M ′′ if and only if σ(x′) = −σ(x′′).


Proof. Denote by ρ′j , 1 6 j 6 m′, the complex line bundles (7.7) correspond-

ing to the characteristic submanifolds of M ′ (or to the facets of P ′), and similarlyfor ρ′′k, 1 6 k 6 m′′, and M ′′. We also denote

c1(ρ′j) = v′j , c1(ρ

′′k) = v′′k , 1 6 j 6 m′, 1 6 k 6 m′′.

The facets of the polytope P ′ # P ′′ are of three types: n facets arising from theidentifications of facets meeting at x′ and x′′, (m′ − n) facets coming from P ′,and (m′′ − n) facets coming from P ′′. We denote the corresponding line bundlesover M ′#M ′′ by ξi, ξ

′j and ξ′′k , respectively (they correspond to the columns of the

characteristic matrix (9.3)). Consider their first Chern classes in H2(M ′#M ′′):

wi = c1(ξi), w′j = c1(ξ

′j), w′′

k = c1(ξ′′k ),

1 6 i 6 n, n+ 1 6 j 6 m′, n+ 1 6 k 6 m′′.

Now consider the maps p′ : M ′#M ′′ → M ′ and p′′ : M ′#M ′′ → M ′′ pinching oneof the connected summands to a point. We have p′ ∗(ρ′j) = ξ′j for n + 1 6 j 6 m′

and p′′ ∗(ρ′′k) = ξ′′k for n+ 1 6 k 6 m′′. Relations (7.17) in the cohomology ring ofM ′#M ′′ take the form

wi = −λ′i,n+1w′n+1 − · · · − λ′i,m′w′

m′ − λ′′i,n+1w′′n+1 − · · · − λ′′i,m′′w′′

m′′ .

It follows that

(9.4) wi = p′ ∗v′i + p′′ ∗v′′i , 1 6 i 6 n.

Since the first n facets of P ′ # P ′′ do not intersect, it follows that w1 · · ·wn = 0 inH2n(M ′#M ′′), hence

(p′ ∗v′1 + p′′ ∗v′′1 ) · · · (p′ ∗v′n + p′′ ∗v′′n) = p′ ∗(v′1 · · · v′n) + p′′ ∗(v′′1 · · · v′′n) = 0.

For any choice of an orientation with the corresponding fundamental class〈M ′#M ′′〉 ∈ H2n(M

′#M ′′), we obtain

v′1 · · · v′n(p′∗〈M ′#M ′′〉

)+ v′′1 · · · v′′n

(p′′∗〈M ′#M ′′〉

)= 0.

An orientation of M ′#M ′′ is compatible with the orientations of M ′ and M ′′ ifand only if p′∗〈M ′#M ′′〉 = 〈M ′〉 and p′′∗〈M ′#M ′′〉 = 〈M ′′〉. Substituting this to theidentity above, we obtain σ(x′) + σ(x′′) = 0.

Proposition 9.1.12. Let M ′ = M(P ′, Λ′′) and M ′′ = M(P ′′, Λ′′) be two om-nioriented quasitoric manifolds, and assume that σ(x′) = −σ(x′′). Then the stablycomplex structure defined on the equivariant connected sum M ′#x′,x′′M ′′ by thecharacteristic matrix (9.3) and the compatible orientation is equivalent to the sumof the canonical stably complex structures on M ′ and M ′′. In particular, the corre-sponding complex bordism classes satisfy

[M ′#M ′′] = [M ′] + [M ′′].

Proof. The connected sum of the two canonical stably complex structures onM ′ and M ′′ is defined by the isomorphism

(9.5) T (M ′#M ′′)⊕ R2(m′+m′′−n) ∼=−→ p′ ∗(ρ′1 ⊕ · · · ⊕ ρ′m′)⊕ p′′ ∗(ρ′′1 ⊕ · · · ⊕ ρ′′m′′)

(see Construction D.3.9). We have p′ ∗(ρ′j) = ξ′j for n+1 6 j 6 m′ and p′′ ∗(ρ′′k) = ξ′′kfor n + 1 6 k 6 m′′. Furthermore, we claim that p′ ∗(ρ′i) ⊕ p′′ ∗(ρ′′i ) = ξi ⊕ C for1 6 i 6 n. Indeed, relations (7.17) for M ′ imply

p′ ∗(ρ′i) = (ξ′n+1)−λ′

i,n+1 ⊗ · · · ⊗ (ξ′m′)−λ′

i,m′ , 1 6 i 6 n,


and the same relations for M ′′ imply

p′′ ∗(ρ′′i ) = (ξ′′n+1)−λ′′

i,n+1 ⊗ · · · ⊗ (ξ′′m′′ )−λ′′

i,m′′ , 1 6 i 6 n.

The line bundle ξ′j over M ′#M ′′ has a section whose zero set is precisely the char-

acteristic submanifold corresponding to the facet F ′j ⊂ P ′#P ′′, for n+1 6 j 6 m′.

There is an analogous property of the bundles ξ′′k , for n + 1 6 k 6 m′′. Now,since the facets F ′

j and F ′′k do not intersect in P ′ # P ′′ for any j and k, the bun-

dle p′ ∗(ρ′i) ⊕ p′′ ∗(ρ′′i ) has a nowhere vanishing section, for 1 6 i 6 n. Therefore,p′ ∗(ρ′i) ⊕ p′′ ∗(ρ′′i ) = η ⊕ C for some line bundle η. By comparing the first Chernclasses and using (9.4), we obtain η = ξi, as needed.

Then the stably complex structure (9.5) takes the form

T (M ′#M ′′)⊕ R2(m′+m′′−n) ∼=−→∼=−→ ξ1 ⊕ · · · ⊕ ξn ⊕ ξ′n+1 ⊕ · · · ⊕ ξ′m′ ⊕ ξ′′n+1 ⊕ · · · ⊕ ξ′′m′′ ⊕ Cn.

This differs by a trivial summand Cn from the stably complex structure defined bymatrix (9.3).

Corollary 9.1.13. The complex bordism class [M ′#M ′′] does not depend onthe choice of initial vertices and the ordering of facets of P ′ and P ′′.

The relationship between the equivariant connected sum M ′#x′,x′′M ′′ of om-nioriented quasitoric manifolds and the standard connected sum M ′ #M ′′ of ori-ented (or stably complex) manifolds is now clear: the two operations produce thesame manifold if and only if σ(x′) = −σ(x′′). Otherwise the equivariant connectedsum gives M ′ # M ′′ or M ′ # M ′′ depending on the choice of orientation. Thisimplies that the equivariant connected sum cannot always be used to obtain thesum of bordism classes. If the sign of every vertex of P is positive, for example,then it is impossible to obtain the bordism class 2[M ] directly from M #M . Thisis the case when M is a toric manifold.

Proof of the main result. We start with an example.

Example 9.1.14. Consider the standard cube In with the orientation inducedfrom Rn. The quasitoric manifold over In corresponding to the characteristic n×2n-matrix (I| −I) (where I is the unit n× n-matrix) is the product (CP 1)n with thestandard complex structure. It represents a nontrivial complex bordism class, andthe signs of all vertices of the cube are positive.

On the other hand, we can consider the omnioriented quasitoric manifoldover In corresponding to the matrix (I|I). It is easy to see that the correspond-ing stably complex structure on (CP 1)n ∼= (S2)n is the product of n copies ofthe trivial structure on CP 1 from Example D.3.1. We denote this omniorientedquasitoric manifold by S. The bordism class [S] is zero, and the sign of a vertex(ε1, . . . , εn) ∈ In where εi = 0 or 1 is given by

σ(ε1, . . . , εn) = (−1)ε1 · · · (−1)εn .So adjacent vertices of In have opposite signs.

We are now in a position to prove the next key lemma which emphasises animportant principle; however unsuitable a quasitoric manifold M may be for theformation of connected sums, a good alternative representative always exists withinthe complex bordism class [M ].


Lemma 9.1.15. Let M be an omnioriented quasitoric manifold of dimension> 2 over a polytope P . Then there exists an omnioriented M ′ over a polytope P ′

such that [M ′] = [M ] and P ′ has at least two vertices of opposite sign.

Proof. Suppose that x is the initial vertex of P . Let S be the omniorientedproduct of 2-spheres of Example 9.1.14, with initial vertex w = (0, . . . , 0).

If σ(x) = −1, define M ′ to be S#w,xM over P ′ = In#w,xP . Then [M ′] = [M ],because S bounds. Moreover, there is a pair of adjacent vertices of In which surviveunder formation of connected sum In #w,x P (because n > 1). These two verticeshave opposite signs, as sought.

If σ(x) = −1, we make the same construction using the opposite orientation ofIn (and therefore of S). Since −S also bounds, the same conclusions hold.

We may now complete the proof of the main result.

Theorem 9.1.16 ([58]). In dimensions > 2, every complex bordism class con-tains a quasitoric manifold, necessarily connected, whose stably complex structureis induced by an omniorientation, and is therefore compatible with the torus action.

Proof. Consider bordism classes [M1] and [M2] in ΩUn , represented by om-nioriented quasitoric manifolds over polytopes P1 and P2 respectively. It thensuffices to construct a quasitoric manifold M such that [M ] = [M1]+ [M2], becauseTheorem 9.1.9 gives an additive basis of ΩUn represented by quasitoric manifolds.

Firstly, we follow Lemma 9.1.15 and replace M2 by M ′2 over P ′

2 = In # P2.Then we choose the initial vertex of P ′

2 so as to ensure that it has the oppositesign to the initial vertex of P1, thereby guaranteeing the construction of M1#M

′2

over P1 #P ′2. The resulting omniorientation defines the required bordism class, by

Proposition 9.1.12 and Lemma 9.1.15.

Combining Theorem 9.1.16 with Proposition 7.3.12 and the quadratic descrip-tion (6.3) of the moment-angle manifold ZP leads to another interesting conclusionrelating toric topology to complex cobordism:

Theorem 9.1.17. Every complex bordism class may be represented by a stablycomplex manifold obtained as the quotient of a free torus action on a completeintersection of real quadrics.

One further deduction from Theorem 9.1.16 is the following result of Ray:

Theorem 9.1.18 ([275]). Every complex bordism class contains a representa-tive whose stable tangent bundle is a sum of line bundles.

Examples. Here we consider some examples of 4-dimensional quasitoric man-ifolds (i.e. n = 2) illustrating the constructions of this section.

Example 9.1.19. When n = 2, the complex bordism class [CP 2] of the stan-dard complex structure of Example 7.3.21 is an additive generator of the groupΩU4∼= Z2, with c2(CP 2) = 3 and all signs of the vertices of the quotient 2-simplex

∆2 being positive.

The question then arises of representing the bordism class 2[CP 2] by an omnior-iented quasitoric manifold M . We cannot expect to use CP 2#CP 2 for M , becauseno vertices of sign −1 are available in ∆2, as required by Lemma 9.1.11. Moreover,M must satisfy c2(M) = 6, by additivity, so the quotient polytope P has 6 or more


vertices (see Exercise 7.3.37). It follows that P cannot be ∆2 # ∆2, which is asquare! So we proceed to appealing to Lemma 9.1.15, and replace the second copyof CP 2 by the omnioriented quasitoric manifold (−S)#CP 2 over P ′ = I2#∆2. Ofcourse (−S)#CP 2 is bordant to CP 2, and P ′ is a pentagon. These observationslead naturally to our second example:

Example 9.1.20. The quasitoric manifold M = CP 2# (−S)#CP 2 representsthe bordism class 2[CP 2], and its quotient is polytope is ∆2 # I2 # ∆2, which isa hexagon. Fig. 9.1 illustrates the procedure diagrammatically, in terms of char-acteristic functions and orientations. Every vertex of the hexagon has sign 1, soM admits an equivariant almost complex structure by Theorem 7.3.23; in fact itcoincides with the manifold from Exercise 7.3.34.

(−1,−1)

(0,1)

(1,0)

(0,1)

(1,0)

(1,0)

(0,1)

(1,0)

(0,1)

(−1,−1)

(−1,−1)

(0,1) (1,0)

(1,0) (0,1)

(−1,−1)

# # =

Figure 9.1. Equivariant connected sum CP 2# (−S) #CP 2.

Our third example shows a related 4-dimensional situation in which the con-nected sum of the quotient polytopes does support a compatible orientation.

Example 9.1.21. Let CP 2denote the quasitoric manifold obtained by revertingthe standard orientation of CP 2 (equivalently, reverting the orientation of the stan-

dard simplex ∆2). Every vertex has sign −1, and we may construct CP 2#CP 2 asan omnioriented quasitoric manifold over ∆2 #∆2. The corresponding characteris-tic functions and orientations are shown in Fig. 9.2. Of course [CP 2] = −[CP 2]. So

(−1,−1)

(0,1)

(1,0)

(0,1)

(1,0)

(−1,−1)

(−1,−1)

(1,0)

(0,1)

(−1,−1)

# =

Figure 9.2. Equivariant connected sum CP 2#CP 2.

[CP 2]+[CP 2] = 0 in ΩU4 , and the manifold CP 2#CP 2bounds by Proposition 9.1.12.

A situation similar to that of Example 9.1.20 arises in higher dimensions, whenwe consider the problem of representing complex bordism classes by toric manifolds.For any such V , the top Chern number cn[V ] coincides with the Euler characteristic,and is therefore equal to the number of vertices of the quotient polytope P .

9.2. THE UNIVERSAL TORIC GENUS 365

Suppose that toric manifolds V1 and V2 are of dimension > 4, and have quotientpolytopes P1 and P2 respectively. Then cn[V1] = f0(P1) and cn[V2] = f0(P2), yetf0(P1 # P2) = f0(P1) + f0(P2) − 2, where f0(·) denotes the number of vertices.Since cn([V1] + [V2]) = cn[V1] + cn[V2], no omnioriented quasitoric manifold overP1 # P2 can represent [V1] + [V2] (see Exercise 7.3.37). This objection vanishes forP1 # In # P2, because it enjoys additional 2n − 2 vertices with opposite signs.

Exercises.

9.1.22. Construct an effective action of a j-dimensional torus T j on a Milnorhypersurface Hij and a representation of T j in C(i+1)(j+1) such that the compo-

sition Hij → CP i × CP j → CP (i+1)(j+1)−1 with the Segre embedding becomesequivariant. Describe the fixed points of this action.

9.1.23. A torus T i+j−1 cannot act effectively with isolated fixed points on aMilnor hypersurface Hij with i > 1. (Hint: use Theorem 7.4.35 and other resultsof Section 7.4.)

9.1.24. Show that the Milnor hypersurface H11 is isomorphic (as a complexmanifold) to the Hirzebruch surface F1 from Example 5.1.8. In particular, H11 isnot homeomorphic to F0 = CP 1 × CP 1 (see Exercise 5.1.13).

9.1.25. Show by comparing the cohomology rings thatH1j is not homeomorphicto CP 1 × CP j−1. On the other hand, the two manifolds are complex bordant byExercise D.5.14.

9.1.26. The procedure described in Example 9.1.20 allows one to constructan almost complex quasitoric manifold M (with all signs positive) representingthe sum of cobordism classes [V1] + [V2] of any two projective toric manifolds ofreal dimension 4. Describe a similar procedure giving almost complex quasitoricrepresentatives for [V1] + [V2] in dimension 6. (Hint: modify the intermediate zero-cobordant manifold S.) What about higher dimensions?

9.2. The universal toric genus

A theory of equivariant genera for stably complex manifolds equipped withcompatible actions of a torus T k was developed in [59]. This theory focuses onthe notion of universal toric genus Φ, defined on stably complex T k-manifolds andtaking values in the complex cobordism ring U∗(BT k) of the classifying space. Theconstruction of Φ goes back to the works of tom Dieck, Krichever and Loffler fromthe 1970s. The universal toric genus Φ is an equivariant analogue of the universalHirzebruch genus (Example E.3.5) corresponding to the identity homomorphismfrom the complex cobordism ring ΩU to itself.

Here is an idea behind the construction of Φ; details are provided below. Westart by defining a composite transformation of T k-equivariant cohomology functors

(9.6) ΦX : U∗Tk(X)

ν−→ MU ∗Tk(X)

α−→ U∗(ET k ×Tk X).

Here U∗Tk(X) (respectively MU ∗

Tk(X)) denotes the geometric (respectively the

homotopic) T k-equivariant complex cobordism ring of a T k-manifold X , andU∗(ET k ×Tk X) denotes the ordinary complex cobordism of the Borel construc-tion. (Note that the geometric and homotopical versions of equivariant cobordismare different, because of the lack of equivariant transversality.)


By restricting (9.6) to the case X = pt we get a homomorphism of ΩU -modules

Φ : ΩU :Tk → ΩU [[u1, . . . , uk]]

from the geometric T k-(co)bordism ring U∗Tk(pt) = Ω∗

U :Tk = ΩU :Tk

−∗ to the ring

U∗(BT k) = ΩU [[u1, . . . , uk]]. Here uj is the cobordism Chern class cU1 (ηj) ofthe canonical line bundle (the conjugate of the Hopf bundle) over the jth fac-tor of BT k = (CP∞)k, for 1 6 j 6 k. We refer to Φ as the universal toric genus.

It assigns to a bordism class [M, cT ] ∈ ΩU :Tk

2n of a 2n-dimensional stably complexT k-manifold M the ‘cobordism class’ of the map ET k ×Tk M → BT k. The valueΦ(M) is a power series in u1, . . . , uk with coefficients in ΩU and constant term [M ].

We now proceed to providing the details of the construction. All our T k-spacesX have homotopy type of cell complexes. It is often important to take account ofbasepoints, in which case we insist that they be fixed by T k. If X itself does nothave a T k-fixed point, then a disjoint fixed basepoint can be added; the result isdenoted by X+.

Homotopic equivariant cobordism. The homotopic version of equivariantcobordism is defined via the Thom T k-spectrum MUTk , whose spaces are indexedby the inclusion poset of complex representations V of T k (of complex dimen-sion |V |). Each MUTk(V ) is the Thom T k-space of the universal |V |-dimensionalcomplex T k-equivariant vector bundle γV over BUTk(V ), and each spectrum mapΣ2(|W |−|V |)MUTk(V ) → MUTk(W ) is induced by the inclusion V ⊂ W of a T k-submodule. The homotopic T k-equivariant complex cobordism group MUnTk(X) of

a pointed T k-space X is defined by stabilising the pointed T k-homotopy sets:

MUnTk(X) = lim−→

[Σ2|V |−n(X+),MUTk(V )

]Tk ,

The details of this construction can be found in [214, Chapters XXV–XXVIII].Applying the Borel construction to γV yields a complex |V |-dimensional bundle

ET k ×Tk γV over ET k ×Tk BUTk(V ), whose Thom space is ET k+ ∧Tk MUTk(V ).

The classifying map for the bundle ET k ×Tk γV induces a map of Thom spacesET k+ ∧Tk MUTk(V ) → MU(|V |). Now consider a T k-map Σ2|V |−n(X+) →MUTk(V ) representing a homotopic cobordism class in MUnTk(X). By applyingthe Borel construction and composing with the classifying map above, we obtain acomposite map of Thom spaces

Σ2|V |−n(ET k ×Tk X)+ −→ ET k+ ∧Tk MUTk(V )→MU(|V |).This construction is homotopy invariant, so we get a map

[Σ2|V |−n(X+),MUTk(V )

]Tk −→

[Σ2|V |−n(ET k ×Tk X)+,MU(|V |)

].

Furthermore, it preserves stabilisation and therefore yields the transformation

α : MU∗Tk(X) −→ U∗(ET k ×Tk X),

which is multiplicative and preserves Thom classes [310, Proposition 1.2].The construction of α may be also interpreted using the homomorphism

MU∗Tk(X) → MU∗

Tk(ETk × X) induced by the T k-projection ET k × X → X ;

since T k acts freely on ET k ×X , the target may be replaced by U∗(ET k ×Tk X).Moreover, α is an isomorphism whenever X is compact and T k acts freely.

Remark. According to the result of Loffler [214, Chapter XXVII], α is thehomomorphism of completion with respect to the augmentation ideal in MU∗

Tk(X).


Geometric equivariant cobordism. The geometric version of equivariantcobordism can be defined naturally by providing an equivariant version of Quillen’sgeometric approach [273] to complex cobordism via complex oriented maps (seeConstruction D.3.3). However, this approach relies on normal complex structures,whereas many of our examples present themselves most readily in terms of tangen-tial information. In the non-equivariant situation, the two forms of data are, ofcourse, interchangeable; but the same does not hold equivariantly. This fact wasoften ignored in early literature, and appears only to have been made explicit in1995, by Comezana [214, Chapter XXVIII, §3]. As we shall see below, tangentialstructures may be converted to normal, but the procedure is not reversible.

We recall from Construction D.3.3 that elements in the cobordism groupU−d(X) of a manifold X can be represented by stably tangentially complex bundlesπ : E → X with d-dimensional fibre F , i.e. by those π for which the bundle TF (E)of tangents along the fibre is equipped with a stably complex structure cT (π).

If π is T k-equivariant bundle, then it is stably tangentially complex as a T k-equivariant bundle when cT (π) is also T k-equivariant. The notions of equivariantequivalence and equivariant cobordism apply to such bundles accordingly.

The geometric T k-equivariant complex cobordism group U−dTk (X) consists of

equivariant cobordism classes of d-dimensional stably tangentially complex T k-equivariant bundles over X . If X = pt , then we may identify both F and E withsome d-dimensional smooth T k-manifold M , and TF (E) with its tangent bundleT (M). So cT (π) reduces to a T k-equivariant stably tangentially complex struc-

ture cT on M , and its cobordism class belongs to the group Ω−dU :Tk = U−d

Tk (pt) (the

cobordism group of point). The bordism group of point is given by ΩU :Tk

d = Ω−dU :Tk .

The direct sums ΩU :Tk

= ΩU :Tk

∗ =⊕

dΩU :Tk

d and ΩU :Tk =⊕

dΩdU :Tk are the geo-

metric T k-equivariant bordism and cobordism rings respectively, and U∗Tk(X) is a

graded ΩU :Tk–module under cartesian product. Furthermore, U∗Tk( ·) is functorial

with respect to pullback along smooth T k-maps Y → X .

Proposition 9.2.1. Given any smooth compact T k-manifold X, there arecanonical homomorphisms

ν : U−dTk (X)→MU−d

Tk (X), d > 0.

Proof. The idea is to convert the tangential structure used in the definitionof geometric cobordism group U−d

Tk (X) to the normal structure required for thePontryagin–Thom collapse map in the homotopical approach.

Let π in Ω−dU :Tk(X) denote the cobordism class of a stably tangentially complex

T k-bundle π : E → X . Choose a T k-equivariant embedding i : E → V into a com-plex T k-representation space V (see Theorem B.3.2) and consider the embedding(π, i) : E → X × V . It is a map of vector bundles over X which is T k-equivariantwith respect to the diagonal action on X×V . There is an equivariant isomorphismc : τF (E) ⊕ ν(π, i) → V = E × V of bundles over E, where ν(π, i) is the normalbundle of (π, i). Now combine c with the stably complex structure isomorphism

cT (π) : TF (E)⊕ R2l−d → ξ to obtain an equivariant isomorphism

(9.7) W ⊕ ν(π, i) −→ ξ⊥ ⊕ V ⊕ R2l−d,

where W = ξ⊥⊕ξ is a T k-decomposition for some complex representation space W .If d is even, (9.7) determines a complex T k-structure on an equivariant stabil-

isation of ν(π, i); if d is odd, a further summand R must be added. For notational


convenience, assume the former, and write d = 2n. We compose (9.7) with the

classifying map ξ⊥ ⊕ V ⊕ Cl−n → γR, where R is a T k-representation of complexdimension |R| = |V |+ |W |−n and then pass to the Thom spaces to get a sequenceof T k-equivariant maps

Σ2|W | Th(ν(π, i)) −→ Th(ξ⊥ ⊕ V ⊕ R2l−d) −→ Th(γR) =MUTk(R).

We compose this with the Pontryagin–Thom collapse map on ν(π, i) to obtain

f(π) : Σ2|V |+2|W |X+ = Σ2|R|−dX+ −→MUTk(R).

If π and π′ are equivalent, then f(π) and f(π′) differ only by suspension; if theyare cobordant, then f(π) and f(π′) are stably T k-homotopic. So we define ν(π) tobe the T k-homotopy class of f(π), as an element of MU−d

Tk (X).

The linearity of ν follows immediately from the fact that addition in U−dTk (X)

is induced by disjoint union.

The proof of Proposition 9.2.1 also shows that ν factors through the geometriccobordism group of stably normally complex T k-manifolds over X .

The universal toric genus. For any smooth compact T k-manifold X , wedefine the homomorphism

ΦX : U∗Tk(X)

ν−→ MU ∗Tk(X)

α−→ U∗(ET k ×Tk X).

Definition 9.2.2. The homomorphism

Φ : ΩU :Tk −→ ΩU [[u1, . . . , uk]]

corresponding to the case X = pt above is called the universal toric genus.

The genus Φ is a multiplicative cobordism invariant of stably complex T k-manifolds, and it takes values in the ring ΩU [[u1, . . . , uk]]. As such it is an equi-variant extension of Hirzebruch’s original notion of genus (see Appendix E.3), andis closely related to the theory of formal group laws. We explore this relation andstudy other equivariant genera in the next sections.

Remark. By the result of Hanke [149] and Loffler [196, (3.1) Satz], whenX = pt , both homomorphisms ν and α are monic; therefore so is Φ.

On the other hand, there are two important reasons why ν cannot be epic.Firstly, it is defined on stably tangential structures by converting them into stablynormal information; this procedure cannot be reversed equivariantly, because theformer are stabilised only by trivial representations of T k, whereas the latter arestabilised by arbitrary representations V . Secondly, homotopical equivariant cobor-dism groups are periodic, and each T k-representation W gives rise to an invertible

Euler class e(W ) in MU2|W |

Tk (pt), while UdTk(pt) = ΩdU :Tk is zero for positive d; thisphenomenon exemplifies the failure of equivariant transversality.

In geometric terms, the universal toric genus Φ assigns to a geometric bor-

dism class [M, cT ] ∈ ΩU :Tk

d of a d-dimensional stably complex T k-manifold M the‘cobordism class’ of the map ET k ×Tk M → BT k. Since both ET k ×Tk M andBT k are infinite-dimensional, one needs to use their finite approximations to definethe cobordism class Φ(M) purely in terms of stably complex structures. Here is aconceptual way to make this precise.


Proposition 9.2.3. Let [M ] ∈ ΩU :Tk be a geometric equivariant cobordismclass represented by a d-dimensional T k-manifold M . Then

Φ(M) = (id×Tk π) !1,

where

(id×Tk π) ! : U∗(ET k ×Tk M) −→ U∗−d(ET k ×Tk pt) = U∗−d(BT k)

is the Gysin homomorphism in cobordism induced by the projection π : M → pt .

Proof. Choose an equivariant embedding M → V into a complex representa-tion space. Then the projection id×Tk π : ET k ×Tk M → BT k factorises as

ET k ×Tk Mi−→ ET k ×Tk V −→ BT k.

We can approximate ET k by the products (S2q+1)k with the diagonal action of T k.Then the above sequence is approximated by the appropriate factorisations ofsmooth bundles over finite-dimensional manifolds,

(9.8) (S2q+1)k ×Tk Miq−→ (S2q+1)k ×Tk V −→ (CP q)k.

This defines a complex orientation for the map (S2q+1)k×TkM → (CP q)k (see Con-struction D.3.3) and therefore a complex cobordism class in U−d((CP q)k), which wedenote by Φq(M). By definition of the Gysin homomorphism (Construction D.3.6),Φq(M) = (idq ×Tk π) !1, where idq is the identity map of (S2q+1)k. As q increases,the classes Φq(M) form an inverse system, whose limit is Φ(M) ∈ U−d(BT k). Inparticular, Φ(M) = (id×Tk π) !1.

Coefficients of the expansion of Φ(M). Recall that the cobordism ringU∗(CP∞) is isomorphic to ΩU [[u]], where u = cU1 (η) ∈ U2(CP∞) is the genera-tor represented geometrically by a codimension-one complex projective subspaceCP∞−1 ⊂ CP∞ (viewed as the direct limit of inclusions CPN−1 ⊂ CPN ).

A basis for the ΩU -module ΩU [[u1, . . . , uk]] = U∗(BT k) in dimension 2|ω| isgiven by the monomials uω = uω1

1 · · ·uωk

k , where ω ranges over nonnegative integralvectors (ω1, . . . , ωk), and |ω| =∑j ωj. A monomial uω is represented geometrically

by a k-fold product of complex projective subspaces of codimension (ω1, . . . , ωk) in(CP∞)k. If we write

(9.9) Φ(M) =∑

ω

gω(M)uω

in ΩU [[u1, . . . , uk]], then the coefficients gω(M) lie in Ω−2(|ω|+n)U . We shall describe

their representatives geometrically as universal operations on M . The cobordismclass gω(M) will be represented by the total space Gω(M) of a bundle with fibreM over the product Bω = Bω1 × · · · × Bωk

, where each Bωiis the bounded flag

manifold (Section 7.7), albeit with the stably complex structure representing zeroin ΩU2ωi

and therefore not equivalent to the standard complex structure on BFωi.

We start by describing stably complex structures on BFn.

Construction 9.2.4. We denote by ξn the ‘tautological’ line bundle over thebounded flag manifold BFn, whose fibre over U ∈ BFn is the first space U1. Werecall from Proposition 7.8.6 that BFn has a structure of a Bott tower in whicheach stage Bk = BF k is the projectivisation CP (ξk−1 ⊕ C). We shall denote thepullback of ξi to the top stage BFn by the same symbol ξi, so that we have n linebundles ξ1, . . . , ξn over BFn.


Using the matrix A corresponding to the Bott manifold BFn (described inExample 7.8.5), we identify BFn with the quotient of the product of 3-spheres

(9.10) (S3)n = (z1, . . . , z2n) ∈ C2n : |zk|2 + |zk+n|2 = 1, 1 6 k 6 nby the action of T n given by

(9.11) (z1, . . . , z2n) 7→ (t1z1, t−11 t2z2, . . . , t

−1n−1tnzn, t1zn+1, t2zn+2, . . . , tnz2n)

(see the proof of Theorem 7.8.7). The manifold BFn is a complex algebraic variety(a toric manifold). The corresponding stably complex structure can be describedby viewing BFn as a quasitoric manifold and applying Theorem 7.3.14, which gives

(9.12) T (BFn)⊕ Cn ∼= ρ1 ⊕ · · · ⊕ ρm,where the ρi are the line bundles (7.7) corresponding to characteristic submanifolds.We have c1(ρi) = vi ∈ H2(BFn), 1 6 i 6 2n, the canonical ring generatorsof the cohomology ring of BFn viewed as a quasitoric manifold. On the otherhand, we have the cohomology ring generators uk = c1(ξk), 1 6 k 6 n, for theBott manifold BFn. The two sets are related by the identities uk = −vk+n (seeExercise 7.8.34). Then identities (7.17) in H∗(BFn) imply that v1 = −u1 andvk = uk−1 − uk, or equivalently, ρ1 = ξ1 and ρk = ξk−1 ξk (where we droppedthe sign of tensor product of line bundles), for 2 6 k 6 n. The stably complexstructure (9.12) therefore becomes

(9.13) T (BFn)⊕ Cn ∼= ξ1 ⊕ ξ1ξ2 ⊕ · · · ⊕ ξn−1ξn ⊕ ξ1 ⊕ ξ2 ⊕ · · · ⊕ ξn(note that when n = 1 we obtain the standard isomorphism T CP 1⊕C ∼= η⊕ η, asξ1 = η is the tautological line bundle).

We shall change the stably complex structure on BFn so that the resultingbordism class in ΩU2n will be zero. To see that this is possible, we regard BFn as asphere bundle over BFn−1 rather that the complex projectivisationCP (ξn−1⊕C). Ifa stably complex structure cT on BFn restricts to a trivial stably complex structureon each fibre S2 (see Example D.3.1), then cT extends over the associated 3-diskbundle, so it is cobordant to zero.

So we need to change the stably complex structure (9.13) so that the new struc-ture restricts to a trivial one on each fibre S2. We decompose (S3)n as (S3)n−1×S3

and T n as T n−1 × T 1, then T 1 acts trivially on (S3)n−1 by (9.11), and we obtain

(9.14) BFn = (S3)n/T n =((S3)n−1 × (S3/T 1)

)/T n−1 = BFn−1 ×Tn−1 S2.

Here T 1 acts on S3 diagonally, so the stably complex structure on S2 is the standardstructure of CP 1 (see Example 7.3.16).

Now change the torus action (9.11) to the following:

(9.15) (z1, . . . , z2n) 7→ (t1z1, t−11 t2z2, . . . , t

−1n−1tnzn, t

−11 zn+1, t

−12 zn+2, . . . , t

−1n z2n).

Then decomposition (9.14) is still valid, but now T 1 acts on S3 antidiagonally,so the stably complex structure on S2 is trivial, as needed. The resulting stablycomplex structure on BFn is given by the isomorphism

(9.16) T (BFn)⊕ R2n ∼= ξ1 ⊕ ξ1ξ2 ⊕ · · · ⊕ ξn−1ξn ⊕ ξ1 ⊕ ξ2 ⊕ · · · ⊕ ξn,and its cobordism class in ΩU2n is zero.


Definition 9.2.5. We denote by Bn the manifold BFn with zero-cobordantstably complex structure (9.16). The ‘tautological’ line bundle ξn is classified by amap Bn → CP∞ and therefore defines a bordism class βn ∈ U2n(CP∞). We alsoset β0 = 1. The set of bordism classes βn : n > 0 is called Ray’s basis of theΩU -module U∗(CP

∞).

Proposition 9.2.6 ([275]). The bordism classes βn : n > 0 form a basis ofthe free ΩU -module U∗(CP∞) which is dual to the basis uk : k > 0 of the ΩU -module U∗(CP∞) = ΩU [[u]]. Here u = cU1 (η) is the cobordism first Chern class ofthe canonical line bundle over CP∞ (represented by CP∞−1 ⊂ CP∞).

Proof. Since [Bn] = 0 in ΩU , the bordism class βn lies in the reduced bordism

module U2n(CP∞) for n > 0. Therefore, to show that βn : n > 0 and uk : k > 0are dual bases it is enough to verify the property

u βn = βn−1

(the -product is defined in Construction D.3.4). The bordism class u βn isobtained by making the map Bn → CP∞ transverse to the zero section CP∞−1

of CP∞ = MU(1) or, equivalently, by restricting the map Bn → CP∞ to thezero set of a transverse section of the line bundle ξn. As is clear from the proofof Proposition 7.7.7, the zero set of a transverse section of ξn = ρ01 is obtained bysetting z1 = 0 in (9.10) and (9.15), which gives precisely Bn−1.

For a nonnegative integer vector ω = (ω1, . . . , ωk), define the manifold Bω =Bω1 × · · · ×Bωk

and the corresponding product bordism class βω ∈ U2|ω|(BTk).

Corollary 9.2.7. The set βω is a basis of the free ΩU -module U∗(BTk);

this basis is dual to the basis uω of U∗(BT k) = ΩU [[u1, . . . , uk]].

Definition 9.2.8. Let M be a tangentially stably complex T k-manifold M .Let Tω be the torus Tω1 × · · · × Tωk and (S3)ω the product (S3)ω1 × · · · × (S3)ωk ,on which Tω acts coordinatewise by (9.15). Define the manifold

Gω(M) = (S3)ω ×Tω M,

where Tω acts on M via the representation

(t1,1, . . . , t1,ω1 ; . . . ; tk,1, . . . , tk,ωk) 7−→ (t1,ω1 , . . . , tk,ωk

).

The stably complex structure on Gω(M) is induced by the tangential structures onthe base and fibre of the bundle M → Gω(M)→ Bω.

Theorem 9.2.9 ([59]). The manifold Gω(M) represents the bordism class of

the coefficient gω(M) ∈ Ω−2(|ω|+n)U of (9.9). In particular, the constant term of

Φ(M) ∈ ΩU [[u1, . . . , uk]] is [M ] ∈ Ω−2nU .

Proof. By Corollary 9.2.7, the coefficient gω(M) is identified with the Kro-necker product 〈Φ(M), βω〉 (see Construction D.3.4). In terms of (9.8), it is repre-sented on the pullback of the diagram

Bω −→ (CP q)k ←− (S2q+1)k ×Tk M

for suitably large q, and therefore on the pullback of the diagram

Bω −→ BT k ←− ET k ×Tk M

of direct limits. The latter pullback is exactly Gω(M).


Remark. There is also a similar description of the coefficients in the expansionof ΦX(π) for the transformation (9.6) in the case when U∗(ET k×Tk X) is a finitelygenerated free U∗(BT k)-module, see [59, Theorem 3.15].

Exercises.

9.2.10. Proposition 9.2.3 can be generalised to the following description of ho-momorphism ΦX given by (9.6). Let π ∈ U−d

Tk (X) be a geometric cobordism class

represented by a T k-equivariant bundle π : E → X . Then

ΦX(π) = (1 ×Tk π) !1,

where

(1 ×Tk π) ! : U∗(ET k ×Tk E) −→ U∗−d(ET k ×Tk X)

is the Gysin homomorphism in cobordism.

9.2.11. The original approach of [275] to define a zero-cobordant stably tangentstructure on BFn is as follows. One can identify BFn with the sphere bundleS(ξn−1 ⊕R) (rather than with CP (ξn−1 ⊕ C)). Let πn : S(ξn−1 ⊕ R)→ BFn−1 bethe projection; show that the tangent bundle of BFn satisfies

T BFn ⊕ R ∼= π∗n(T BFn−1 ⊕ ξn−1 ⊕ R).

By identifying π∗nξn−1 with ξn−1 (as bundles over BFn), we obtain inductively

T BFn ⊕ R ∼= ξ1 ⊕ ξ2 ⊕ · · · ⊕ ξn−1 ⊕ R.

Show that this stably complex structure is equivalent to that of (9.16). (Hint:calculate the total Chern classes.)

9.2.12. There is a canonical T k-action onGω(M) and Bω makingGω(M)→ Bωinto a T k-equivariant bundle.

9.3. Equivariant genera, rigidity and fibre multiplicativity

Recall that a genus is a multiplicative cobordism invariant of stably complexmanifolds, i.e. a ring homomorphism ϕ : ΩU → R to a commutative ring with unit.We only consider genera taking values in torsion-free rings R; such ϕ are uniquelydetermined by series f(x) = x + · · · ∈ R ⊗ Q[[x]] via Hirzebruch’s correspondence(Construction E.3.1).

Historically, equivariant extensions of genera were first considered by Atiyahand Hirzebruch [10], who established the rigidity property of the χy-genus and

A-genus of S1-manifolds. The origins of these concepts lie in the Atiyah–Bottfixed point formula [9], which also acted as a catalyst for the development of equi-variant index theory. This development culminated in the celebrated result ofTaubes [302] establishing the rigidity of the Ochanine–Witten elliptic genus onspin S1-manifolds.

Here we develop an approach to equivariant genera and rigidity based solely onthe complex cobordism theory. It allows us to define an equivariant extension andthe appropriate concept of rigidity for an arbitrary Hirzebruch genus, and agreeswith the classical index-theoretical approach when the genus is the index of anelliptic complex.

9.3. EQUIVARIANT GENERA, RIGIDITY AND FIBRE MULTIPLICATIVITY 373

9.3.1. Equivariant genera. Our definition of an equivariant genus uses auniversal transformation of cohomology theories, studied in [47]:

Construction 9.3.1 (Chern–Dold character). Consider multiplicative trans-formations of cohomology theories

h : U∗(X)→ H∗(X ;ΩU ⊗Q).

Such h is determined uniquely by a series h(u) ∈ H2(CP∞;ΩU ⊗Q) = ΩU⊗Q[[x]],where u = cU1 (η) ∈ U2(CP∞) and x = cH1 (η) ∈ H2(CP∞).

The Chern–Dold character is the unique multiplicative transformation

chU : U∗(X)→ H∗(X ;ΩU ⊗Q)

which reduces to the canonical inclusion ΩU → ΩU ⊗Q in the case X = pt .

Proposition 9.3.2. The Chern–Dold character satisfies

chU (u) = fU (x),

where fU (x) is the exponential of the formal group law FU of geometric cobordisms.

Proof. Since chU acts identically on ΩU , it follows that

(9.17) chU FU (v1, v2) = FU (chU (v1), chU (v2))

for any v1, v2 ∈ U2(CP∞). Let f(x) denote the series chU (u) ∈ ΩU ⊗ Q[[x]]. Letvi = cU1 (ξi) and xi = cH1 (ξi) for i = 1, 2. Then

chU FU (v1, v2) = chU(cU1 (ξ1 ⊗ ξ2)

)= f

(cH1 (ξ1 ⊗ ξ2)

)= f(x1 + x2),

chU (v1) = f(x1), chU (v2) = f(x2).

Substituting these expressions in (9.17) we get f(x1+x2) = FU (f(x1), f(x2)), whichmeans that f(x) is the exponential of FU .

Given a Hirzebruch genus ϕ : Ω → R ⊗ Q corresponding to f(x) ∈ R⊗ Q[[x]],we define a multiplicative transformation

hϕ : U∗(X)

chU−→ H∗(X ;ΩU ⊗Q)ϕ−→ H∗(X ;R⊗Q)

where the second homomorphism acts by ϕ on the coefficients only. In the caseX = BT k we obtain a homomorphism

hϕ : ΩU [[u1, . . . , uk]]→ R ⊗Q[[x1, . . . , xk]]

which acts on the coefficients as ϕ and sends ui to f(xi) for 1 6 i 6 k.

Definition 9.3.3 (equivariant genus). The T k-equivariant extension of ϕ isthe ring homomorphism

ϕT : ΩU :Tk → R⊗Q[[x1, . . . , xk]]

defined as the composition hϕ · Φ with the universal toric genus.

Remark. In the definition of equivariant genus from [59], the generator uiwas sent to xi instead of f(xi). Although this does not affect the notion of rigid-ity defined below, Definition 9.3.3 is a more natural extension of Krichever’s andAtiyah–Hirzebruch’s approaches.


Rigidity. Krichever [188], [189] considers rational valued genera ϕ : ΩU → Q,and equivariant extensions ϕK : ΩU :Tk → K∗(BT k)⊗Q. The equivariant genus ϕK

is related to ours ϕT via a natural transformation U∗(X) → K∗(X) ⊗ Q definedby ϕ, as explained below. If ϕ(M) can be realised as the index ind(E) of anelliptic complex E of complex vector bundles over M (see e.g. [161]), then for

any T k-manifold M this index has a natural T k-equivariant extension indTk

(E)which is an element of the complex representation ring RU (T

k), and hence in itscompletion K0(BT k). Krichever’s interpretation of rigidity is to require that ϕK

should lie the subring of constants Q for every M . In the case of an index, thisamounts to insisting that the corresponding T k-representation is always trivial, andtherefore conforms to Atiyah and Hirzebruch’s original notion [10].

The composition of hϕ : Ueven(X)→ Heven(X ;Q) with the inverse of the Chern

character ch: K0(X)⊗Q∼=→ Heven(X ;Q) gives the transformation

hKϕ = ch−1 · hϕ : Ueven(X)→ K0(X)⊗Q,

considered by Krichever in [188]. The transformation Uodd(X) → K1(X) ⊗ Q isdefined similarly; together they give a transformation hKϕ : U∗(X)→ K∗(X)⊗Q.

Remark. In the case of the Todd genus td: ΩU → Z this construction givesthe Conner–Floyd transformation U∗ → K∗ described after Example E.3.6.

Krichever [189] referred to a genus ϕ : ΩU → Q as rigid if the composition

ϕK : ΩU :TkΦ−→ U∗(BT k)

hKϕ−→ K0(BT k)⊗Q

belongs to the subring Q ⊂ K0(BT k) ⊗ Q, i.e. satisfies ϕK(M) = ϕ(M) forany [M ] ∈ ΩU :Tk . Definition 9.3.3 of equivariant genera based on the notion ofthe universal toric genus leads naturally to the following version of rigidity, whichsubsumes the approaches of Atiyah–Hirzebruch and Krichever:

Definition 9.3.4. A genus ϕ : ΩU → R is T k-rigid on a stably complex T k-manifold M whenever ϕT : ΩU :Tk → R ⊗ Q[[u1, . . . , uk]] satisfies ϕT (M) = ϕ(M);if this holds for every M , then ϕ is T k-rigid.

Since the Chern character ch: K0(X)⊗Q→ Heven(X ;Q) is an isomorphism,a rational genus ϕ is T k-rigid in the sense of Definition 9.3.4 if and only if it is rigidin the sense of Krichever (and therefore in the original index-theoretical sense ofAtiyah–Hirzebruch if ϕ is an index).

In Section 9.5 we shall describe how toric methods can be applied to establishthe rigidity property for several fundamental Hirzebruch genera. Now we consideranother important property of genera.

Fibre multiplicativity. The following definition extends that of Hirze-bruch [161, Chapter 4] for the oriented case. It applies to fibre bundles of the

form M → E ×G M π→ B, where M and B are closed, connected and stably tan-gentially complex, G is a compact Lie group of positive rank whose action preservesthe stably complex structure on M , and E → B is a principal G-bundle. In thesecircumstances, the bundle π is stably tangentially complex, and N = E ×G Minherits a canonical stably complex structure.

Definition 9.3.5. A genus ϕ : ΩU → R is fibre multiplicative with respect tothe stably complex manifold M whenever ϕ(N) = ϕ(M)ϕ(B) for any such bundleπ with fibre M ; if this holds for every M , then ϕ is fibre multiplicative.

9.4. ISOLATED FIXED POINTS: LOCALISATION FORMULAE 375

For rational genera in the oriented category, Ochanine [249, Proposition 1]proved that rigidity is equivalent to fibre multiplicativity (see also [161, Chap-ter 4]). In the toric case, we have the following stably complex analogue, whose

conclusions are integral. It refers to bundles E ×GM π−→ B of the form requiredby Definition 9.3.5, where G has maximal torus T k with k > 1.

Theorem 9.3.6 ([59]). If the genus ϕ is T k-rigid on M , then it is fibre mul-tiplicative with respect to M for bundles whose structure group G has the propertythat U∗(BG) is torsion-free.

On the other hand, if ϕ is fibre multiplicative with respect to a stably tangentiallycomplex T k-manifold M , then it is T k-rigid on M .

Proof. Let ϕ be T k-rigid on M , and consider the pullback squares

E ×GM f ′

−−−−→ EG×GM i′←−−−− ET k ×Tk M

π

y πG

y πTk

y

Bf−−−−→ BG

i←−−−− BT k,

where πG is universal, i is induced by inclusion, and f classifies π. By Propo-sition 9.2.3 and commutativity of the right square, πG! 1 = [M ] · 1 + β, where

1 ∈ U0(EG ×G M) and β ∈ U−2n(BG). By commutativity of the left square,π !1 = [M ] · 1 + f∗β in U−2n(B). Applying the Gysin homomorphism associatedwith the augmentation map εB : B → pt yields

(9.18) [E ×GM ] = εB! π !1 = [M ][B] + εB! f∗β

in ΩU2(n+b), where dimB = 2b; so ϕ(E ×G M) = ϕ(M)ϕ(B) + ϕ(εB! f∗β). More-

over, i∗β =∑

|ω|>0 gω(M)uω in U−2n(BT k), so ϕ(i∗β) = 0 because ϕ is T k-rigid.

The assumptions on G ensure that i∗ is injective, which implies that ϕ(β) = 0 inU∗(BG)⊗ϕ R. Fibre multiplicativity then follows from (9.18).

Conversely, suppose that ϕ is fibre multiplicative with respect to M , and con-sider the manifold Gω(M) of Theorem 9.2.9. By Definition 9.2.8, it is the totalspace of the bundle (S3)ω ×TωM → Bω, which has structure group T k; thereforeϕ(Gω(M)) = 0, because Bω bounds for every |ω| > 0. So ϕ is T k-rigid on M .

Remark. We may define ϕ to be G-rigid when ϕ(β) = 0, as in the proof ofTheorem 9.3.6. It follows that T -rigidity implies G-rigidity for any G such thatΩ∗U (BG) is torsion-free.

Example 9.3.7. The signature (or the L-genus, see Example E.3.6.2) is fibremultiplicative over any simply connected base [161, Chapter 4], and so is rigid.

9.4. Isolated fixed points: localisation formulae

In this section we focus on stably tangentially complex T k-manifolds (M2n, cT )for which the fixed points p are isolated; in other words, the fixed point set MT isfinite. We proceed by deducing a localisation formula for Φ(M) in terms of fixedpoint data. We give several illustrative examples, and describe the consequencesfor certain non-equivariant genera and their T k-equivariant extensions.

Localisation theorems in equivariant generalised cohomology theories appearin the works of tom Dieck [310], Quillen [273], Krichever [188], Kawakubo [180],


and elsewhere. We prove our Theorem 9.4.1 by interpreting their results in the caseof isolated fixed points, and identifying the signs explicitly.

Each integer vector w = (w1, . . . , wk) ∈ Zk determines a line bundle

ηw = ηw11 ⊗ · · · ⊗ ηwk

k

over BT k = (CP∞)k, where ηj is the canonical line bundle over the jth factor. Let

[w ,u ] = cU1 (ηw )

denote the cobordism first Chern class of ηw . It is given by the power series

FU (u1, . . . , u1︸︷︷︸w1

, . . . , uk, . . . , uk︸︷︷︸wk

) ∈ U2(BT k),

where FU (u1, . . . , uk) is the iterated substitution FU (· · ·FU (FU (u1, u2), u3), . . . , uk)in the formal group law of geometric cobordisms, see Section E.2. Modulo decom-posables we have that

(9.19) [w ,u ] ≡ w1u1 + · · ·+ wkuk,

and it is convenient to rewrite the right hand side as a scalar product 〈w ,u〉.Let p be an isolated fixed point for the T k-action on M . We recall from Sec-

tion D.6 that the weights w j(p) ∈ Zk and the sign σ(p) = ±1 are defined, and referto w j(p), σ(p) : 1 6 j 6 n, p ∈MT as the fixed point data of (M, cT ).

Theorem 9.4.1 (localisation formula). For any stably tangentially complex 2n-dimensional T k-manifold M with isolated fixed points MT , the equation

(9.20) Φ(M) =∑

p∈MT

σ(p)n∏

j=1

1

[wj(p),u ]

is satisfied in U−2n(BT k).

Remark. The summands on the right hand side of (9.20) formally belong tothe localised ring S−1U∗(BT k) where S is the set of equivariant Euler classes ofnontrivial representation of T k.

Proof. Choose an equivariant embedding i : M → V into a complex N -dimensional representation space V and consider the commutative diagram

(9.21)

MT rM−−−−→ M

iT

yyi

V TrV−−−−→ V

where V T ⊂ V is the T k-fixed subspace, rM and rV denote the inclusions of fixedpoints, and iT is the restriction of i to MT .

We restrict (9.21) to a tubular neighbourhood of MT in V , which can beidentified with the total space E of the normal bundle ν = ν(MT → V ):

(9.22)

MT rM−−−−→ E1

iT

yyi

E2rV−−−−→ E


where E1 = ν(rM ) and E2 = ν(iT ). The normal bundle ν decomposes as

(9.23) ν = ν(rM )⊕ r∗M (ν(i)) = ν(rM )⊕ ν(iT )⊕ ζ,where ζ is the ‘excess’ bundle over MT , whose fibres are the non-trivial parts ofthe T k-representations in the fibres of r∗Mν(i). We therefore rewrite (9.22) as

(9.24)

MT rM−−−−→ E1

iT

yyi

E2rV−−−−→ E1 ⊕ E2 ⊕ F,

where F is the total space of ζ. Since all relevant bundles are complex, we haveGysin–Thom isomorphisms (see Construction D.3.6 and Exercise D.3.14)

i ! : U∗(M)

∼=−→ U∗+p(Th(ν(i))

)= U∗+p(V, V \M) = U∗+p(E,E \ E1),

iT! : U∗(MT )

∼=−→ U∗+q(Th(ν(iT ))

)= U∗+q(V T , V T \MT ) = U∗+q(E2, E2 \MT ),

where p = dimV − dimM and q = dimV T − dimMT . Let i1 : E1 → E1 ⊕ E2,i2 : E2 → E1⊕E2 and k : E1⊕E2 → E be the inclusion maps. Then for x ∈ U∗(M),

r∗V i !x = i∗2k∗k !i1!x = i∗2

(e(ν(k)) · i1!x

).

Since i∗2(ν(k)) = π∗(ζ), where π : E2 → M is the projection, the last term abovecan be written as

π∗(e(ζ)) · i∗2i1!x = π∗(e(ζ)) · iT! r∗Mx by Proposition D.3.7 (e)

= iT!(iT∗π∗(e(ζ)) · r∗Mx

)by Proposition D.3.7 (d)

= iT!(e(ζ) · r∗Mx

).

We therefore obtain

(9.25) r∗V i !x = iT!(e(ζ) · r∗Mx

)

in U∗+p(V T , V T \MT ).

Given a T k-equivariant map f : M → N we denote by f its ‘Borelification’, i.e.the map ET k ×Tk M → ET k ×Tk N . Applying this procedure to (9.21) we obtaina commutative diagram

(9.26)

BT k ×MT rM−−−−→ ET k ×Tk M

iT

yyi

BT k × V T rV−−−−→ ET k ×Tk V.

Using the finite-dimensional approximation of the above diagram (as in the proof ofProposition 9.2.3) we can view it as a diagram of proper maps of smooth manifoldsand therefore apply Gysin homomorphisms in cobordism. By analogy with (9.25)we obtain for x ∈ U∗(ET k ×Tk M),

(9.27) r∗V i !x = iT!(e(ζ) · r∗Mx

),

where ζ is the ‘excess’ bundle over BT k ×MT defined similarly to (9.23).


Similarly, by considering the diagram

0 0

jTy

yj

V TrV−−−−→ V

we obtain for y ∈ U∗(BT k),

(9.28) r∗V j !y = jT!(e(ζV ) · y

),

where ζV is the nontrivial part of the T k-representation V , i.e. V = V T ⊕ ζV . Letπ : M → pt and πT : MT → pt be the projections. Then i ! = j ! · π !, i

T! = jT! · πT! .

Substituting y = π !x in (9.28) we obtain

r∗V i !x = jT!(e(ζV ) · π !x

)

Comparing this to (9.27) and using the fact that

jT! : U∗(BT k)→ U∗+r(Th(ν(jT ))) = U∗+r(ΣrBT k)

is an isomorphism (here r = dim V T ), we finally obtain

e(ζV ) · π !x = πT!(e(ζ) · r∗Mx

).

Now set x = 1. Then π !1 = Φ(M) by Proposition 9.2.3 and r∗M1 = 1. We get

(9.29) e(ζV ) · Φ(M) = πT!(e(ζ)

).

This formula is valid without restrictions on the fixed point set. Now, ifMT is finite,

then πT! (e(ζ)) =∑p∈MT e(ζ|p). Recall that ζ is defined from the decomposition ν =

ν(rM )⊕ν(iT )⊕ζ, in which ν|p = ν(MT → V )|p can be identified with V and ν(iT )|pcan be identified with V T (because p is isolated). Since V = V T ⊕ ζV , it follows

that e(ζV ) = e(ν(rM )|p)e(ζ|p) for any p ∈MT . We therefore can rewrite (9.29) as

Φ(M) =∑

p∈MT

1

e(ν(rM )|p

) .

It remains to note that ν(rM )|p = ν(p → M) is the tangential T k-representationTpM , so ν(rM )|p is the bundle ET k ×Tk TpM over BT k, whose Euler class is

e(ν(rM )|p

)= σ(p)

n∏

j=1

[w j(p),u ]

by the definition of sign σ(p) and weights w j(p).

Remark. If the structure cT is almost complex, then σ(x) = 1 for all fixedpoints x, and (9.20) reduces to Krichever’s formula [189, (2.7)]).

The left-hand side of (9.20) lies in ΩU [[u1, . . . , uk]], whereas the right-handside appears to belong to an appropriate localisation. It follows that all terms ofnegative degree must cancel, thereby imposing substantial restrictions on the fixedpoint data. These may be made explicit by rewriting (9.19) as

(9.30) [w , tu ] ≡ (w1u1 + · · ·+ wkuk) t mod (t2)


in ΩU [[u1, . . . , uk, t]], and then defining the power series

(9.31)∑

l

cf l tl = tnΦ(M)(tu) =

∑

p∈MT

σ(p)

n∏

j=1

t

[w j(p), tu ]

over the localisation of ΩU [[u1, . . . , uk]].

Proposition 9.4.2. The coefficients cf l are zero for 0 6 l < n, and satisfy

cfn+m =∑

|ω|=m

gω(M)uω

for m > 0; in particular, cfn = [M ].

Proof. Combine the definitions of cf l in (9.31) and gω in (9.9).

Remark. The equations cf l = 0 for 0 6 l < n are the T k-analogues of theConner–Floyd relations for Zp-actions [246, Appendix 4]; the extra equation cfn =[M ] provides an expression for the cobordism class of M in terms of fixed pointdata. This is important because, according to Theorem 9.1.16, every element ofΩU may be represented by a stably tangentially complex T k-manifold with isolatedfixed points. We explore on this in the next section.

Now let ϕ : ΩU → R be a genus taking values in a torsion-free ring R, with thecorresponding series f(x) = x+ · · · ∈ R⊗Q[[x]]. We may adapt Theorem 9.4.1 toexpress ϕ(M) in terms of fixed point data. The resulting formula is much simpler,because the formal group law ϕFU may be linearised over R⊗Q:

Proposition 9.4.3. Let ϕ : ΩU → R be a genus with torsion-free R, and letM be a stably tangentially complex 2n-dimensional T k-manifold with isolated fixedpoints MT . Then the equivariant genus ϕT (M) = ϕ(M) + · · · is given by

(9.32) ϕT (M) =∑

p∈MT

σ(p)

n∏

j=1

1

f(〈wj(p), x〉

) ,

where 〈w, x〉 = w1x1 + · · ·+ wkxk for w = (w1, . . . , wk).

Proof. By Theorem E.3.3, f(x) is the exponential series of the formal grouplaw ϕFU , i.e. ϕFU (u1, u2) = f(f−1(u1) + f−1(u2)), and therefore hϕFU (u1, u2) =f(x1 + x2). An iterated application of this formula gives hϕ([w j(p),u ]) =f(〈w j(p), x 〉). Since ϕT = hϕ · Φ, the result follows from Theorem 9.4.1.

Example 9.4.4. The augmentation genus ε : ΩU → Z corresponds to the seriesf(x) = x; it vanishes on any M2n with n > 0. Formula (9.32) then gives

(9.33)∑

p∈MT

σ(p)

n∏

j=1

1

〈w j(p), x 〉= 0.

Let M = CPn on which T n+1 acts homogeneous coordinatewise. There are n + 1fixed points p0, . . . , pn, each having a single nonzero coordinate. So the weightvector w j(pk) is ej − ek for 0 6 j 6 n, j 6= k, and every σ(pk) is positive;thus (9.33) reduces to the classical identity

n∑

k=0

∏

06j6n

j 6=k

1

xj − xk= 0 .


Example 9.4.5. Consider the S1-action preserving the standard complex struc-ture on CP 1. It has two fixed points, both with signs 1 and weights 1 and −1,respectively (see Example D.6.3). Theorem 9.4.1 gives the following expression forthe universal toric genus:

Φ(CP 1) =1

u+

1

u

in U−2(CP∞) where u = [−1, u] is the inverse series in the formal group law ofgeometric cobordisms.

By Theorem 9.4.3, a genus ϕ : ΩU → R is rigid on CP 1 only if its definingseries f(x) satisfies the equation

(9.34)1

f(x)+

1

f(−x) = c

in R ⊗Q[[x]]. The general analytic solution is of the form

(9.35) f(x) =x

q(x2) + cx/2, where q(0) = 1

(an exercise). The Todd genus of Example E.3.6.3 is defined by the series f(x) =1− e−x, and (9.34) is satisfied with c = 1. So td is T -rigid on CP 1, and

q(x2) =x

2· e

x/2 + e−x/2

ex/2 − e−x/2

in Q[[x]]. In fact td is fibre multiplicative with respect to CP 1 by [160], so rigidityalso follows from Theorem 9.3.6.

We can also consider the S1-action on M = CP 1 with trivial stably complexstructure. It has two fixed points of signs 1 and −1, both with weights 1. The-orem 9.4.1 gives the universal toric genus Φ(M) = 1

u − 1u = 0, which also follows

from the fact that M bounds equivariantly.

Another classical application of the localisation formula is the Atiyah–Hirzebruch formula [10] expressing the χy-genus of a complex S1-manifold interms of the fixed point data. We discuss a generalisation of this formula due toKrichever [188]. It refers to the 2-parameter χa,b-genus corresponding to the series

(9.36) f(x) =eax − ebxaebx − beax ∈ Q[a, b].

The χy-genus (Example E.3.7) corresponds to the parameter values a = y, b = −1.We choose a circle subgroup in T k defined by a primitive vector ν ∈ Zk:

S(ν) = (e2πiν1ϕ, . . . , e2πiνkϕ) ∈ Tk : ϕ ∈ R.

We have MS(ν) =MT for a generic circle S(ν) (see Lemma 7.4.3). The weights ofthe tangential representation of S(ν) at p are 〈w j(p), ν〉, 1 6 j 6 n. If fixed points

MT are isolated and MS(ν) =MT , then

(9.37) 〈w j(p), ν〉 6= 0 for 1 6 j 6 n and any p ∈MT .

We define the index indν p as the number of negative weights at p, i.e.

indν p = #j : 〈w j(p), ν〉 < 0.


Theorem 9.4.6 (generalised Atiyah–Hirzebruch formula [188]). The χa,b-genus is T k-rigid. Furthermore, the χa,b-genus of a stably tangentially complex2n-dimensional T k-manifold M with finite MT is given by

(9.38) χa,b(M) =∑

p∈MT

σ(p)(−a)indν p(−b)n−indν p

for any ν ⊂ Zk satisfying MS(ν) =MT .

Remark. The original formula of Atiyah–Hirzebruch [10] was given for com-plex manifolds. Krichever implicitly assumed manifolds to be almost complex whendeducing his formula, as no signs were mentioned in [188]. However his proof,presented below, automatically extends to the stably complex situation by incor-porating the signs of fixed points.

Also, both Atiyah–Hirzebruch’s and Krichever’s formulae are valid withoutassuming the fixed points to be isolated, see Exercise 9.4.9. We give the formula inthe case of isolated fixed points to emphasise the role of signs.

Proof of Theorem 9.38. We establish the rigidity and prove the formulafor M with isolated fixed points only; the proof in the general case is similar. By

Proposition 9.4.3, to prove the T k-rigidity on M it suffices to prove that χS(ν)a,b is

constant for any S(ν) satisfying MS(ν) =MT . Formula (9.32) gives

(9.39) χS(ν)a,b (M) =

∑

p∈MT

σ(p)

n∏

j=1

aeb〈wj(p),ν〉x − bea〈wj(p),ν〉x

ea〈wj(p),ν〉x − eb〈wj(p),ν〉x.

This expression belongs to Z[a, b][[x]] (that is, it is non-singular at zero) and itsconstant term is χa,b(M). We denote ωj = 〈w j(p), ν〉 and e(a−b)x = q; then we mayrewrite each factor in the product above as

(9.40)aebωjx − beaωjx

eaωjx − ebωjx=a− b qωj

qωj − 1.

Then (9.39) takes the following form:

(9.41) χS(ν)a,b (M) =

∑

p∈MT

σ(p)

n∏

j=1

a− b qωj

qωj − 1.

Now we let q → ∞. Then (9.40) has limit −b if ωj > 0 and limit −a if ωj < 0.Therefore, the limit of (9.41) is

∑

p∈MT

σ(p)(−a)indν p(−b)n−indν p.

Similarly, the limit of (9.41) as q → 0 is∑

p∈MT

σ(p)(−a)n−indν p(−b)indν p.

Therefore, χS(ν)a,b (M) has value for q = 0 as well as for q = ∞. Since it is a finite

Laurent series in q, it must be constant in q, and its value coincides with either ofthe limits above.

Remark. It follows also that the right hand side of (9.38) is independent of ν;the two limits above are taken to each other by substitution ν → −ν.


Exercises.

9.4.7. The general analytic solution of (9.34) is given by (9.35).

9.4.8. The formal group law corresponding to the χa,b-genus is given by

F (u, v) =u+ v + (a+ b)uv

1− ab · uv .

9.4.9. Let M be a stably tangentially complex T k-manifold M with fixed pointset MT . Then

χa,b(M) =∑

F⊂MT

χa,b(F )(−a)indν F (−b)ℓ−indν F

for any ν ⊂ Zk satisfying MS(ν) = MT , where the sum is taken over connectedfixed submanifolds F ⊂ MT , 2ℓ = dimM − dimF , and indν F is the number ofnegative weights of the S(ν)-action in the normal bundle of F .

9.5. Quasitoric manifolds and genera

In the case of quasitoric manifolds, the combinatorial description of signs offixed points and weights obtained in Section 7.3 opens a way to effective calcula-tion of characteristic numbers and Hirzebruch genera using localisation techniques.We illustrate this approach by presenting formulae expressing the χa,b-genus (inparticular, the signature and the Todd genus) of a quasitoric manifold as a sumof contributions depending only on the ‘local combinatorics’ near the vertices ofthe quotient polytope. These formulae were obtained in [256]; they can also bededuced from the results of [206] in the more general context of torus manifolds.

Localisation formulae for genera on quasitoric manifolds can be interpreted asfunctional equations on the series f(x). By resolving these equations for particularexamples of quasitoric manifolds one may derive different ‘universality theorems’for rigid genera. For example, according to a result of Musin [237], the χa,b-genusis universal for T k-rigid genera (this implies that any T k-rigid rational genus isχa,b for some rational parameters a, b). We prove this result as Theorem 9.5.6 byresolving the functional equation coming from localisation formula (9.32) on CP 2

with a nonstandard omniorientation.

We assume given a combinatorial quasitoric pair (P,Λ) (see Definition 7.3.10)and the corresponding omnioriented quasitoric manifold M = M(P,Λ). This fixesa T n-invariant stably complex structure on M and the corresponding bordism class[M ] ∈ ΩU2n, as described in Corollary 7.3.15.

Any fixed point v ∈M is given by the intersection of n characteristic subman-ifolds v =Mj1 ∩ · · ·∩Mjn and corresponds to a vertex of the polytope P , which wealso denote by v. The expressions for the weights w j(v) and the sign σ(v) in termsof the quasitoric pair (P,Λ) are given by Proposition 7.3.17 and Lemma 7.3.18.

In the quasitoric case the condition (9.37) guarantees that the circle S(ν) sat-isfies MT =MS(ν) (an exercise).

Example 9.5.1 (Chern number cn[M ]). The series (9.36) defining the χa,b-genus has limit as a− b→ 0, which can be calculated as follows:

eax − ebxaebx − beax =

1− e(b−a)xae(b−a)x − b =

(a− b)x+ · · ·(a− b)− a(a− b)x+ · · · =

x

1− ax.

9.5. QUASITORIC MANIFOLDS AND GENERA 383

For a = b = −1 we obtain the defining series for the top Chern number cn[M ] (seeExample E.3.6.1). Plugging these values into (9.38) we obtain

cn[M ] =∑

v∈MT

σ(M),

which we already know from Exercise 7.3.37. When all signs are positive, we obtaincn[M ] = χ(M) = f0(P ), i.e. the Euler characteristic of M is equal to the numberof vertices of P .

Example 9.5.2 (signature). Substituting a = 1, b = −1 in (9.36) we obtain theseries tanh(x) defining the L-genus or the signature (see Example E.3.6.2). Being anoriented cobordism invariant, the signature sign(M) does not depend on the stablycomplex structure, i.e. only the global orientation part of the omniorientation dataaffects the signature. The following statement gives a formula for sign(M) whichdepends only on the orientation:

Proposition 9.5.3. For an oriented quasitoric manifold M ,

sign(M) =∑

v∈MT

det(w1(v), . . . , wn(v)

),

where wj(v) are the vectors defined by the conditions

wj(v) = ±wj(v) and(wj(v), ν

)> 0, 1 6 j 6 n.

Proof. Plugging a = 1, b = −1 into (9.38) we obtain

(9.42) sign(M) =∑

v∈MT

(−1)indν(v)σ(v).

Using expression (7.15) for the sign σ(v) we calculate

(−1)indν(v)σ(v) = (−1)indν(v) det(w1(v), . . . ,wn(v)

)= det

(w1(v), . . . , wn(v)

),

which implies the required formula.

If M is a projective toric manifold VP , then σ(v) = 1 for any v and for-mula (9.42) gives

sign(VP ) =∑

v

(−1)indν(v).

Furthermore, in this case the weights w1(v), . . . ,wn(v) are the primitive vectorsalong the edges of P pointing out of v (see Example 7.3.19). It follows that theindex indν(v) coincides with the index defined in the proof of Dehn–Sommervilleequations (Theorem 1.3.4), and we obtain the formula known in toric geometry(see [250, Theorem 3.12]):

sign(VP ) =

n∑

k=0

(−1)khk(P ).

Note that if n is odd then the sum vanishes by the Dehn–Sommerville equations.

Example 9.5.4 (Todd genus). Substituting a = 0, b = −1 in (9.36) we obtainthe series 1− e−x defining the Todd genus (see Example E.3.6.3). We cannot pluga = 0 directly into (9.38), but it is clear from the proof that when a = 0, only


vertices of index 0 contribute (−b)n to the sum. This gives the following formulafor the Todd genus of a quasitoric manifold:

(9.43) td(M) =∑

v : indν(v)=0

σ(v).

When M is projective toric manifold VP , there is only one vertex of index 0.It is the ‘bottom’ vertex, which has all incident edges pointing out (in the notationused in the proof of Theorem 1.3.4). Since σ(v) = 1 for every v ∈ P , formula (9.43)gives td(VP ) = 1, which is well-known (see, e.g. [122, §5.3]).

In the almost complex case we have the following result:

Proposition 9.5.5. If a quasitoric manifold M admits an equivariant almostcomplex structure, then td(M) > 0.

Proof. We choose a compatible omniorientation, so that σ(v) > 0 for any v.Then (9.43) implies td(M) > 0, and we need to show that there is at least one vertexof index 0. Let v be any vertex. Since w1(v), . . . ,wn(v) are linearly independent,we may choose ν so that 〈w j(v), ν〉 > 0 for any j. Then indν v = 0. The resultfollows by observation that td(M) is independent of ν.

A description of td(M) which is independent of ν is outlined in Exercise 9.5.8.

The following result of Musin [236] can be proved by application of localisationformula for quasitoric manifolds:

Theorem 9.5.6. The 2-parameter genus χa,b is universal for T k-rigid genera.In particular, any T k-rigid rational genus is χa,b for some rational parameters a, b.

Proof. The rigidity of χa,b is established by Theorem 9.38. To see that anyT k-rigid genus is χa,b we solve the functional equation arising from the localisationformula for one particular example of T k-manifold.

We consider the quasitoric manifold M = CP 2 with a nonstandard omniorien-tation defined by the characteristic matrix

Λ =

(1 0 10 1 −1

)

It has three fixed points v1, v2, v3, whose corresponding minors Λvi are obtainedby deleting the ith column of Λ. The weights are given by (1, 0), (1, 1),(0,−1), (1, 1) and (0, 1), (1, 0), respectively, see Fig. 9.3. The signs are cal-culated using formula (7.15):

σ(v1) =

∣∣∣∣1 11 0

∣∣∣∣ = −1, σ(v2) =

∣∣∣∣0 1−1 1

∣∣∣∣ = 1, σ(v3) =

∣∣∣∣1 00 1

∣∣∣∣ = 1.

Plugging these data into formula (9.32) we obtain that a genus ϕ is rigid on Monly if its defining series f(x) satisfies the equation

− 1

f(x1)f(x1 + x2)+

1

f(−x2)f(x1 + x2)+

1

f(x1)f(x2)= c.

Interchanging x1 and x2 gives

(9.44) − 1

f(x2)f(x1 + x2)+

1

f(−x1)f(x1 + x2)+

1

f(x2)f(x1)= c ,

9.5. QUASITORIC MANIFOLDS AND GENERA 385

v3 (1, 0) (1, 0) v1

λ2 = (0, 1)

(0, 1)

λ1 = (1, 0)

(0,−1)

v2

(1, 1)

(1, 1)

λ3 = (1,−1)

Figure 9.3. CP 2 with nonstandard omniorientation

and subtraction yields(

1

f(x1)+

1

f(−x1)

)1

f(x1 + x2)=

(1

f(x2)+

1

f(−x2)

)1

f(x1 + x2).

It follows that

1

f(x)+

1

f(−x) = c′ and1

f(−x) = c′ − 1

f(x)

for some constant c′. Substituting in (9.44) gives(

1

f(x1)+

1

f(x2)− c′

)1

f(x1 + x2)=

1

f(x1)f(x2)− c ,

which rearranges to

f(x1 + x2) =f(x1) + f(x2)− c′f(x1)f(x2)

1− cf(x1)f(x2).

So f is the exponential series of the formal group law F (x1, x2) corresponding toχa,b, with c′ = −a− b and c = ab (see Exercise 9.4.8).

Exercises.

9.5.7. If ν ∈ Zn satisfies (9.37), then the circle subgroup S(ν) ⊂ T n acts onthe quasitoric manifold M with isolated fixed points corresponding to the verticesof the quotient polytope P .

9.5.8. Let M =M(P,Λ) be a quasitoric manifold. We realise the dual complexKP as a triangulated sphere with vertices at the unit vectors ai

|ai|, i = 1, . . . ,m.

Then one can define a continuous piecewise smooth map f : Sn−1 → Sn−1 bysending ai

|ai|to λi

|λi|(here λi is the ith column of Λ) and extending the map smoothly

on the spherical simplices corresponding to I ∈ KP . Such an extension is welldefined because the vectors λi : i ∈ I are linearly independent, so one alwayschooses the smallest spherical simplex spanned by them.


Show that td(M) = deg f . (Hint: the number of preimages of ν|ν| ∈ Sn−1 under

f is equal to the number of maximal simplices Iv ∈ K such that all coefficients in thedecomposition of ν via λi : i ∈ Iv are positive; these coefficients are 〈w i(v), ν〉.)

More generally, the Todd genus of a torus manifold can be calculated in thisway as the degree of the corresponding multi-fan, see [153, §3].

9.5.9. Calculate cn[M ], the signature and the Todd genus for quasitoric mani-folds of Example 7.3.21, Example 7.3.22 and Exercise 7.3.34.

APPENDIX A

Commutative and homological algebra

Here we review some basic algebraic notions and results in a way suited fortopological applications. In order to make algebraic constructions compatible withtopological ones we sometimes use a notation which may seem unusual to a readerwith an algebraic background. This in particular concerns the way we treat gradingsand resolutions.

We fix a ground ring k, which is always assumed to be a field or the ring Z ofintegers. In the latter case by a ‘k-vector space of dimension d’ we mean an abeliangroup of rank d.

A.1. Algebras and modules

A k-algebra (or simply algebra) A is a ring which is also a k-vector space, andwhose multiplication A×A→ A is k-bilinear. (The latter condition is void if k = Z,so Z-algebras are ordinary rings.) All our algebras will be commutative and withunit 1, unless explicitly stated otherwise. The basic example is A = k[v1, . . . , vm],the polynomial algebra in m generators, for which we shall often use a shortenednotation k[m].

An algebra A is finitely generated if there are finitely many elements a1, . . . , anof A such that every element of A can be written as a polynomial in a1, . . . , anwith coefficients in k. Therefore, a finitely generated algebra is the quotient of apolynomial algebra by an ideal.

An A-module is a k-vector space M on which A acts linearly, that is, there isa map A×M → M which is k-linear in each argument and satisfies 1m = m and(ab)m = a(bm) for all a, b ∈ A, m ∈M . Any ideal I of A is an A-module. If A = k,then an A-module is a k-vector space.

An A-module M is finitely generated if there exist x1, . . . , xn in M such thatevery element x of M can be written (not necessarily uniquely) as x = a1x1 + · · ·+anxn, ai ∈ A.

An algebra A is Z-graded (or simply graded) if it is represented as a direct sumA =

⊕i∈ZA

i such that Ai ·Aj ⊂ Ai+j . Elements a ∈ Ai are said to be homogeneousof degree i, denoted deg a = i. The set of homogeneous elements of A is denoted byH(A) = ⋃iAi. An ideal I of A is homogeneous if it is generated by homogeneouselements. In most cases our graded algebras will be either nonpositively graded (i.e.Ai = 0 for i > 0) or nonnegatively graded (i.e. Ai = 0 for i < 0); the latter isalso called an N-graded algebra. A nonnegatively graded algebra A is connectedif A0 = k. For a nonnegatively graded algebra A, define the positive ideal byA+ =

⊕i>0A

i; if A is connected and k is a field then A+ is a maximal ideal.

If A is a graded algebra, then an A-module M is graded if M =⊕

i∈ZMi

such that Ai ·M j ⊂ M i+j . An A-module map f : M → N between two graded

387

388 A. COMMUTATIVE AND HOMOLOGICAL ALGEBRA

modules is degree-preserving (or of degree 0) if f(M i) ⊂ N i, and is of degree k iff(M i) ⊂ N i+k for all i.

Graded algebras arising in topology are often graded-commutative (or skew-commutative) rather than commutative in the usual sense. This means that

ab = (−1)ij ba for any a ∈ Ai, b ∈ Aj .

If the characteristic of k is not 2, then the square of an odd-degree element in agraded-commutative algebra is zero. To avoid confusion we double the grading incommutative algebras A; the resulting graded algebras A =

⊕i∈ZA

2i are commu-tative in either sense.

For example, we make the polynomial algebra k[v1, . . . , vm] graded by settingdeg vi = 2. It then becomes a free graded commutative algebra on m generatorsof degree two (free means no relations apart from the graded commutativity). Theexterior algebra Λ[u1, . . . , um] has relations u2i = 0 and uiuj = −ujui. We shallassume deg ui = 1 unless otherwise specified. An exterior algebra is a free gradedcommutative algebra if the characteristic of k is not 2.

Bigraded (i.e. Z ⊕ Z-graded) and multigraded (Zm-graded) algebras A aredefined similarly; their homogeneous elements a ∈ A have bidegree bideg a = (i, j) ∈Z⊕ Z or multidegree mdeg a = i ∈ Zm respectively.

We continue assuming A to be (graded) commutative. The tensor productM ⊗AN of A-modules M and N is the quotient of a free A-module with generatorset M ×N by the submodule generated by all elements of the following types:

(x+ x′, y)− (x, y)− (x′, y), (x, y + y′)− (x, y)− (x, y′),

(ax, y)− a(x, y), (x, ay)− a(x, y),

where x, x′ ∈ M , y, y′ ∈ N , a ∈ A. For each basis element (x, y), its image inM ⊗A N is denoted by x⊗ y.

We shall denote the tensor productM⊗kN of k-vector spaces by simplyM⊗N .For example, if M = N = k[v], then M ⊗N = k[v1, v2].

The tensor product A⊗B of graded-commutative algebras A and B is a gradedcommutative algebra, with the multiplication defined on homogeneous elements by

(a⊗ b) · (a′ ⊗ b′) = (−1)deg bdeg a′aa′ ⊗ bb′.

An A-module F is free if it is isomorphic to a direct sum⊕

i∈I Fi, where eachFi is isomorphic to A as an A-module. If both A and F are graded then everyFi is isomorphic to a j-fold suspension sjA for some j, where sjA is the gradedA-module with (sjA)k = Ak−j . A basis of a free A-module F is a set S of elementsof F such that each x ∈ F can be uniquely written as a finite linear combination ofelements of S with coefficients in A. If A is finitely generated then all bases havethe same cardinality (an exercise), called the rank of F . If S is a basis of a freeA-module F , then for any A-module M a set map S →M extends uniquely to anA-module homomorphism F →M .

A module P is projective if for any epimorphism of modules p : M → N andhomomorphism f : P → N , there is a homomorphism f ′ : P →M such that pf ′ =

A.1. ALGEBRAS AND MODULES 389

f . This is described by the following commutative diagram:

Mp // N // 0

P

f

OO

f ′

``

Equivalently P is projective if it is a direct summand of a free module (an exercise).In particular, free modules are projective.

A sequence of homomorphisms of A-modules

· · · −→M1f1−→M2

f2−→M3f3−→M4 −→ · · ·

is called an exact sequence if Im fi = Ker fi+1 for all i.A chain complex is a sequence C∗ = Ci, ∂i of A-modules Ci and homomor-

phisms ∂i : Ci → Ci−1 such that ∂i∂i+1 = 0. This is usually written as

· · · −→ Ci+1∂i+1−→ Ci

∂i−→ Ci−1 −→ · · ·The condition ∂i∂i+1 = 0 implies that Im ∂i+1 ⊂ Ker∂i. The elements of Ker ∂ arecalled cycles, and the elements of Im ∂ are boundaries. The ith homology group (orhomology module) of C∗ is defined by

Hi(C∗) = Ker ∂i/ Im∂i+1.

A cochain complex is a sequence C∗ = Ci, di of A-modules Ci and homo-morphisms di : Ci → Ci+1 such that didi−1 = 0. This is usually written as

· · · −→ Ci−1 di−1

−→ Cidi−→ Ci+1 −→ · · · .

The elements of Ker d are called cocycles, and the elements of Im d are coboundaries.The ith cohomology group (or cohomology module) of C∗ is defined by

Hi(C∗) = Ker di/ Imdi−1.

A cochain complex may be also viewed as a graded k-vector space C∗ =⊕

i Ci in

which every graded component Ci is an A-module, together with an A-linear mapd : C∗ → C∗ raising the degree by 1 and satisfying the condition d2 = 0.

Note that a chain complex may be turned to a cochain complex by invertingthe grading (i.e. turning the ith graded component into the (−i)th).

A map of cochain complexes is a graded A-module map f : C∗ → D∗ whichcommutes with the differentials. Such a map induces a map in cohomology

f : H(C∗)→ H(D∗), which is also an A-module map.Let f, g : C∗ → D∗ be two maps of cochain complexes. A cochain homotopy

between f and g is a set of maps s = si : Ci → Di−1 satisfying the identities

ds+ sd = f − g(more precisely, di−1si + si+1di = f i − gi). This is described by the followingcommutative diagram

· · · // Ci−1 di−1//

si−1

||②②②②②②②②②

Cidi //

si||②②②②②②②② fi−gi

Ci+1 //

si+1||②②②②②②②②②

· · ·

· · · // Di−1

di−1

// Di

di// Di+1 // · · ·


If there is a cochain homotopy between f and g, then f and g induce the same mapin cohomology (an exercise). A chain homotopy between maps of chain complexesis defined similarly.

A differential graded algebra (a dg-algebra for short) is a graded algebra Atogether with a k-linear map d : A → A, called the differential, which raises thedegree by one, and satisfies the identity d2 = 0 (so that Ai, di is a cochaincomplex) and the Leibniz identity

(A.1) d(a · b) = da · b+ (−1)ia · db for a ∈ Ai, b ∈ A.In order to emphasise the differential, we may display a dg-algebra A as (A, d). Itscohomology H(A, d) = Ker d/ Im d is a graded algebra (an exercise). Differentialgraded algebras whose differential lowers the degree by one are also considered, inwhich case homology is a graded algebra.

A quasi-isomorphism between dg-algebras is a homomorphism f : A→ B which

induces an isomorphism in cohomology, f : H(A)∼=−→ H(B).

Exercises.

A.1.1. If A is a finitely generated algebra, then all bases of a free A-modulehave the same cardinality.

A.1.2. A module is projective if and only if it is a direct summand of a freemodule.

A.1.3. Prove the following extended version of the 5-lemma. Let

C1 //

f1

C2 //

f2

C3 //

f3

C4 //

f4

C5

f5

D1 // D2 // D3 // D4 // D5

be a commutative diagram with exact rows. Then

(a) if f2 and f4 are monomorphisms and f1 is an epimorphism, then f3 is amonomorphism;

(b) if f2 and f4 are epimorphisms and f5 is a monomorphism, then f3 is anepimorphism.

Therefore, is f1, f2, f4, f5 are isomorphisms, then f3 is also an isomorphism.

A.1.4. Cochain homotopic maps between cochain complexes induce the samemaps in cohomology.

A.1.5. The cohomology of a differential graded algebra is a graded algebra.

A.2. Homological theory of graded rings and modules

From now on we assume that A is a commutative finitely generated k-algebrawith unit, graded by nonnegative even numbers (i.e. A =

⊕i>0A

i) and connected

(i.e. A0 = k). The basic example to keep in mind is A = k[m] = k[v1, . . . , vm]with deg vi = 2, however we shall need a greater generality occasionally. We alsoassume that all A-modules M are nonnegatively graded and finitely generated, andall module maps are degree-preserving, unless the contrary is explicitly stated.

A.2. HOMOLOGICAL THEORY OF GRADED RINGS AND MODULES 391

A free (respectively, projective) resolution of M is an exact sequence of A-modules

(A.2) · · · d−→ R−i d−→ · · · d−→ R−1 d−→ R0 →M → 0

in which all R−i are free (respectively, projective) A-modules. A free resolutionexists for every M (an exercise, or see constructions below). The minimal numberp for which there exists a projective resolution (A.2) with R−i = 0 for i > p iscalled the projective (or homological) dimension of the module M ; we shall denoteit by pdimAM or simply pdimM . If such p does not exist, we set pdimM = ∞.The module Mi = Ker[d : R−i+1 → R−i+2] is called the ith syzygy module for M .

If k is a field and A is as above, then an A-module is projective if and only if itis free (see Exercise A.2.15), and we therefore need not to distinguish between freeand projective resolutions in this case.

We can convert a resolution (A.2) into a bigraded k-vector spaceR =⊕

i,j R−i,j

where R−i,j = (R−i)j is the jth graded component of the module R−i, and the(−i, j)th component of d acts as d−i,j : R−i,j → R−i+1,j. We refer to the firstgrading of R as external ; it comes from the indexing of the terms in the resolutionand is therefore nonpositive by our convention. The second, internal, grading of Rcomes from the grading in the modules R−i and is therefore even and nonnegative.The total degree of an element of R is defined as the sum of its external and internaldegrees. We can view A as a bigraded algebra with trivial first grading (i.e. Ai,j = 0for i 6= 0 and A0,j = Aj); then R becomes a bigraded A-module.

If we drop the term M in resolution (A.2), then the resulting cochain complexis exactly R (with respect to its external grading), and we have

H−i,j(R, d) = Kerd−i,j/ Imd−i−1,j = 0 for i > 0,

H0,j(R, d) =M j.

We may view M as a trivial cochain complex 0→M → 0, or as a bigraded modulewith trivial external grading, i.e. M i,j = 0 for i 6= 0 and M0,j = M j. Thenresolution (A.2) can be interpreted as a map of cochain complexes of A-modules:

(A.3)

· · · −−−−→ R−i d−−−−→ · · · d−−−−→ R−1 d−−−−→ R0 −−−−→ 0y

yy

· · · −−−−→ 0 −−−−→ · · · −−−−→ 0 −−−−→ M −−−−→ 0

or simply as a map (R, d)→ (M, 0) inducing an isomorphism in cohomology.The Poincare series of a graded k-vector space V =

⊕i V

i whose gradedcomponents are finite-dimensional is given by

F (V ;λ) =∑

i

(dimk Vi)λi.

Proposition A.2.1. Let (A.2) be a finite free resolution of an A-module, inwhich R−i is a free module of rank qi on generators of degrees d1i, . . . , dqii. Then

F (M ;λ) = F (A;λ)∑

i>0

(−1)i(λd1i + · · ·+ λdqii).

Proof. Since H−i,j(R, d) = 0 for i > 0 and H0,j(R, d) =M j , we obtain∑

i>0

(−1)i dimkR−i,j = dimkM

j


by a basic property of the Euler characteristic. Multiplying by λj and summing upover j we obtain ∑

i>0

(−1)iF (R−i;λ) = F (M ;λ).

Since each R−i is a free A-module, its Poincare series is given by F (R−i;λ) =F (A;λ)(λd1i + · · ·+ λdqii), which implies the required formula.

Construction A.2.2 (minimal resolution). Let k be a field, and let M =⊕i>0M

i be a graded A-module, which is not necessarily finitely generated, but for

which every graded component M i is finite-dimensional as a k-vector space. Thereis the following canonical way to construct a free resolution for M .

Take the lowest degree i in which M i 6= 0 and choose a k-vector space basisin M i. Span an A-submodule M1 by this basis and then take the lowest degree inwhich M 6= M1. In this degree choose a k-vector space basis in the complementof M1, and span a module M2 by this basis and M1. Continuing this process weobtain a system of generators for M which has a finite number of elements in eachdegree, and has the property that images of the generators form a basis of thek-vector space M ⊗A k = M/(A+ ·M). A system of generators of M obtained inthis way is referred to as minimal (or as a minimal basis).

Now choose a minimal generating set in M and span by its elements a free A-module R0

min. Then we have an epimorphism R0min →M . Next we choose a minimal

basis in the kernel of this epimorphism, and span by it a free module R−1min. Then

choose a minimal basis in the kernel of the map R−1min → R0

min, and so on. At the

ith step we choose a minimal basis in the kernel of the map d : R−i+1min → R−i+2

min

constructed in the previous step, and span a free module R−imin by this basis. As a

result we obtain a free resolution of M , which is referred to as minimal. A minimalresolution is unique up to an isomorphism.

Proposition A.2.3. For a minimal resolution of M , the induced maps

R−imin ⊗A k→ R−i+1

min ⊗A k

are zero for i > 1.

Proof. By construction, the map R0min ⊗k A → M ⊗k A is an isomorphism,

which implies that the kernel of the map R0min → M is contained in A+ · R0

min.

Similarly, for each i > 1 the kernel of the map d : R−imin → R−i+1

min is contained in

A+ · R−imin, and therefore the image of the same map is contained in A+ · R−i+1

min .

This implies that the induced maps R−imin ⊗A k→ R−i+1

min ⊗A k are zero.

Remark. If k = Z then the above described inductive procedure still gives aminimal basis for an A-module M , but the kernel of the map d : R0

min → M may

not be contained in A+ ·R0min, and the induced map R−1

min⊗A Z→ R0min ⊗A Z may

be nonzero.

Construction A.2.4 (Koszul resolution). Let A = k[v1, . . . , vm] and M = k

with the A-module structure given by the augmentation map sending each vi tozero. We turn the tensor product

E = Em = Λ[u1, . . . , um]⊗ k[v1, . . . , vm]


into a bigraded differential algebra by setting

(A.4)bideg ui = (−1, 2), bideg vi = (0, 2),

dui = vi, dvi = 0

and requiring d to satisfy the Leibniz identity (A.1). Then (E, d) together with theaugmentation map ε : E → k defines a cochain complex of k[m]-modules

(A.5) 0→ Λm[u1, . . . , um]⊗ k[v1, . . . , vm]d−→ · · ·

d−→ Λ1[u1, . . . , um]⊗ k[v1, . . . , vm]d−→ k[v1, . . . , vm]

ε−→ k→ 0,

where Λi[u1, . . . , um] is the subspace of Λ[u1, . . . , um] generated by monomials oflength i. We shall show that the complex above is an exact sequence, or equivalently,that ε : (E, d)→ [k, 0] is a quasi-isomorphism. There is an obvious inclusion η : k→E such that εη = id. To finish the proof we shall construct a cochain homotopybetween id and ηε, that is, a set of k-linear maps s = s−i,2j : E−i,2j → E−i−1,2jsatisfying the identity

(A.6) ds+ sd = id− ηε.For m = 1 we define the map s1 : E

0,∗1 = k[v]→ E−1,∗

1 by the formula

s1(a0 + a1v + · · ·+ ajvj) = u(a1 + a2v + · · ·+ ajv

j−1).

Then for f = a0 + a1v + · · · + ajvj ∈ E0,∗

1 we have ds1f = f − a0 = f − ηεf

and s1df = 0. On the other hand, for uf ∈ E−1,∗1 we have s1d(uf) = uf and

ds1(uf) = 0. In any case (A.6) holds. Now we may assume by induction thatfor m = k − 1 the required cochain homotopy sk−1 : Ek−1 → Ek−1 is alreadyconstructed. Since Ek = Ek−1 ⊗ E1, εk = εk−1 ⊗ ε1 and ηk = ηk−1 ⊗ η1, a directcalculation shows that the map

sk = sk−1 ⊗ id + ηk−1εk−1 ⊗ s1is a cochain homotopy between id and ηkεk.

Since Λi[u1, . . . , um]⊗ k[m] is a free k[m]-module, (A.5) is a free resolution forthe k[m]-module k. It is known as the Koszul resolution. It can be shown to beminimal (an exercise).

Let (A.2) be a projective resolution of an A-module M , and N is anotherA-module. Applying the functor ⊗AN to (A.3) we obtain a homomorphism ofcochain complexes

(R⊗A N, d)→ (M ⊗A N, 0),which does not induce a cohomology isomorphism in general. The (−i)th gradedcohomology module of the cochain complex

(A.7) · · · → R−i ⊗A N → · · · → R−1 ⊗A N → R0 ⊗A N → 0

is denoted by Tor−iA (M,N). We shall also consider the bigraded A-module

TorA(M,N) =⊕

i,j>0

Tor−i,jA (M,N)

where Tor−i,jA (M,N) is the jth graded component of Tor−iA (M,N)

The following properties of Tor−iA (M,N) are well-known (see e.g. [203]).

Theorem A.2.5.


(a) The module Tor−iA (M,N) does not depend, up to isomorphism, on a choiceof resolution (A.2);

(b) Tor−iA ( · , N) and Tor−iA (M, · ) are covariant functors;

(c) Tor0A(M,N) =M ⊗A N ;

(d) Tor−iA (M,N) ∼= Tor−iA (N,M);(e) An exact sequence of A-modules

0 −→M1 −→M2 −→M3 −→ 0

induces the following long exact sequence:

· · · −→ Tor−iA (M1, N) −→ Tor−iA (M2, N) −→ Tor−iA (M3, N) −→ · · ·· · · −→ Tor−1

A (M1, N) −→ Tor−1A (M2, N) −→ Tor−1

A (M3, N)

−→ Tor0A(M1, N) −→ Tor0A(M2, N) −→ Tor0A(M3, N) −→ 0.

In the case N = k the Tor-modules can be read from a minimal resolution ofM as follows:

Proposition A.2.6. Let k be a field, and let (A.2) be a minimal resolution ofan A-module M . Then

Tor−iA (M,k) ∼= R−imin ⊗A k,

dimk Tor−iA (M,k) = rankR−i

min.

Proof. Indeed, the differentials in the cochain complex

· · · −→ R−imin ⊗A k −→ · · · −→ R−1

min ⊗A k −→ R0min ⊗A k −→ 0

are all trivial by Proposition A.2.3.

Corollary A.2.7. Let k be a field, and let M be a A-module. Then

pdimM = maxi : Tor−iA (M,k) 6= 0

.

Corollary A.2.8. If k is a field, then pdimM 6 m for any k[v1, . . . , vm]-module M .

Proof. By the previous corollary and Theorem A.2.5 (d),

pdimM = maxi : Tor−i

k[m](M,k) 6= 0= max

i : Tor−i

k[m](k,M) 6= 0.

Using the Koszul resolution for the k[m]-module k we obtain

Tor−ik[m](k,M) = H−i

(Λ[u1, . . . , um]⊗ k[m]⊗k[m] M,d

)

= H−i(Λ[u1, . . . , um]⊗M,d

).

Therefore,

pdimM = maxi : Tor−i

k[m](k,M) 6= 06 max

i : Λi[u1, . . . , um]⊗M 6= 0

= m.

Example A.2.9. Let A = k[v1, . . . , vm] and M = N = k. By the minimalityof the Koszul resolution,

Tork[v1,...,vm](k,k) = Λ[u1, . . . , um]

and pdimA k = m.


We note that in general there is no canonical way to define a multiplica-tion in TorA(M,N), even if both M and N are A-algebras rather than just A-modules. However, in the particular case when A = k[m], M is an algebra witha unit and N = k there is the following canonical way to define a product inTorA(M,N), extending the previous example. We consider the differential bigradedalgebra (Λ[u1, . . . , um]⊗M,d) whose bigrading and differential are defined similarlyto (A.4):

(A.8)bideg ui = (−1, 2), bideg x = (0, deg x) for x ∈M,

dui = vi · 1, dx = 0

(here vi · 1 is the element of M obtained by applying vi ∈ k[m] to 1 ∈ M , and weidentify ui with ui ⊗ 1 and x with 1 ⊗ x for simplicity). Using the fact that thecohomology of a differential graded algebra is a graded algebra we obtain:

Lemma A.2.10. Let M be a graded k[v1, . . . , vm]-algebra. Then Tork[m](M,k)is a bigraded k-algebra whose product is defined via the isomorphism

Tork[v1,...,vm]

(M,k

) ∼= H(Λ[u1, . . . , um]⊗M,d

).

Proof. Using the Koszul resolution in the definition of Tork[m](k,M) andTheorem A.2.5 (d) we calculate

Tork[m](M,k) ∼= Tork[m](k,M)

= H(Λ[u1, . . . , um]⊗ k[m]⊗k[m] M,d

) ∼= H(Λ[u1, . . . , um]⊗M,d

).

The algebra(Λ[u1, . . . , um] ⊗ M,d

)is known as the Koszul algebra (or the

Koszul complex ) of M .

Remark. Lemma A.2.10 holds also in the case when M does not have unit(e.g., when it is a graded ideal in k[m]). In this case formula (A.8) for the differentialneeds to be modified as follows:

d(uix) = vi · x, dx = 0 for x ∈M.

We finish this discussion of Tork[m](M,k) by mentioning an important conjec-ture of commutative homological algebra.

Conjecture A.2.11 (Horrocks, see [46, p. 453]). Let M be a gradedk[v1, . . . , vm]-module such that dimkM <∞, where k is a field. Then

dimk Tor−ik[v1,...,vm](M,k) >

(mi

).

It is sometimes formulated in a weaker form:

Conjecture A.2.12 (weak Horrocks’ Conjecture). Let M be a gradedk[v1, . . . , vm]-module such that dimkM <∞, where k is a field. Then

dimk Tork[v1,...,vm](M,k) > 2m.

If the algebra A is not necessarily commutative, then TorA(M,N) is defined fora right A-module M and a left A-module N in the same way as above. However,in this case TorA(M,N) is no longer an A-module, and is just a k-vector space. Ifboth M and N are A-bimodules, then TorA(M,N) is an A-bimodule itself.

The construction of Tor can also be extended to the case of differential gradedmodules and algebras, see Section B.4.


In the standard notation adopted in the algebraic literature, the modules in aresolution (A.2) are numbered by nonnegative rather than nonpositive integers:

· · · d−→ Rid−→ · · · d−→ R1 d−→ R0 →M → 0.

In this notation, the ith Tor-module is denoted by TorAi (M,N), i > 0 (note

that (A.7) becomes a chain complex, and TorA∗ (M,N) is its homology). Therefore,the two notations are related by

Tor−iA (M,N) = TorAi (M,N).

Applying the functor HomA( , N) to (A.3) (with R−i replaced by Ri) we obtainthe cochain complex

0→ HomA(R0, N)→ HomA(R

1, N)→ · · · → HomA(Ri, N)→ · · · .

Its ith cohomology module is denoted by ExtiA(M,N).The properties of the functor Ext are similar to those given by Theorem A.2.5

for Tor, with the exception of (d):

Theorem A.2.13.

(a) The module ExtiA(M,N) does not depend, up to isomorphism, on a choiceof resolution (A.2);

(b) ExtiA( · , N) is a contravariant functor, and ExtiA(M, · ) is a covariantfunctor;

(c) Ext0A(M,N) = HomA(M,N);(d) An exact sequence of A-modules

0 −→M1 −→M2 −→M3 −→ 0


0 −→ Ext0A(M3, N) −→ Ext0A(M2, N) −→ Ext0A(M1, N)

−→ Ext1A(M3, N) −→ Ext1A(M2, N) −→ Ext1A(M1, N) −→ · · ·· · · −→ ExtiA(M3, N) −→ ExtiA(M2, N) −→ ExtiA(M1, N) −→ · · · ;

(e) An exact sequence of A-modules

0 −→ N1 −→ N2 −→ N3 −→ 0


0 −→ Ext0A(M,N1) −→ Ext0A(M,N2) −→ Ext0A(M,N3)

−→ Ext1A(M,N1) −→ Ext1A(M,N2) −→ Ext1A(M,N3) −→ · · ·· · · −→ ExtiA(M,N1) −→ ExtiA(M,N2) −→ ExtiA(M,N3) −→ · · · .

Exercises.

A.2.14. Show that a free resolution exists for every A-module M . (Hint: usethe fact that every module is the quotient of a free module.)

A.2.15. If k is a field and A = k[m], then every projective graded A-moduleis free (hint: see [203, Lemma VII.6.2]). This is also true in the ungraded case,but is much harder to prove (a theorem of Quillen and Suslin, settling the famousproblem of Serre). More generally, if A is a finitely generated nonnegatively gradedcommutative connected algebra over a field k, then every projective A-module is

A.3. COHEN–MACAULAY ALGEBRAS 397

free (see [106, Theorem A3.2]). Give an example of a projective module over a ringwhich is not free.

A.2.16. The Koszul resolution is minimal.

A.3. Regular sequences and Cohen–Macaulay algebras

Cohen–Macaulay algebras and modules play an important role in commutativealgebra, algebraic geometry and combinatorics. Their definition uses the notionof a regular sequence (see Definition A.3.1 below), which also plays an importantrole in algebraic topology, namely in the construction of new cohomology theories(see [191] and Appendix, Section E.3). In the case of finitely generated algebrasover a field k, an algebra is Cohen–Macaulay if and only if it is a free module offinite rank over a polynomial subalgebra.

Here we consider nonnegatively evenly graded finitely generated commutativeconnected algebras A over a field k and finitely generated nonnegatively gradedA-modules M (the case k = Z requires extra care, and is treated separately insome particular cases in the main chapters of the book). The positive part A+ isthe unique homogeneous maximal ideal of A, and the results we discuss here areparallel to those from the homological theory of Noetherian local rings (we referto [45, Chapters 1–2] or [106, Chapter 19] for the details).

Given a sequence of elements t = (t1, . . . , tk) of A, we denote by A/t thequotient of A by the ideal generated by t , and denote by M/tM the quotient of Mby the submodule t1M + · · · + tkM . An element t ∈ A is called a zero divisor onM if tx = 0 for some nonzero x ∈M . An element t ∈ A is not a zero divisor on Mif and only if the map M

t−→M given by multiplication by t is injective.

Definition A.3.1. Let M be an A-module. A homogeneous sequence t =(t1, . . . , tk) ∈ H(A+) is called an M -regular sequence if ti+1 is not a zero divisor onM/(t1M + · · ·+ tiM) for 0 6 i < k. We often refer to A-regular sequences simplyas regular.

The importance of regular sequences in homological algebra builds on the funda-mental fact that an exact sequence of modules remains exact after taking quotientsby a regular sequence:

Proposition A.3.2. Assume given an exact sequence of A-modules:

· · · −→ Sifi−→ Si−1 fi−1−→ · · · f1−→ S0 f0−→ M → 0

If t is an M -regular and Si-regular sequence for all i > 0, then

· · · → Si/tSifi−→ Si−1/tSi−1

fi−1−→ · · · f1−→ S0/tS0 f0−→ M/tM → 0

is an exact sequence of A/t-modules.

Proof. Using induction we reduce the statement to the case when t consistsof a single element t. Since

Si/tSi = Si ⊗A (A/t),

and ⊗A(A/t) is a right exact functor, it is enough to verify exactness of thequotient sequence starting from the term S1/tS1.


Consider the following fragment of the quotient sequence (i > 1):

Si+1/tSi+1 fi+1−→ Si/tSifi−→ Si−1/tSi−1 fi−1−→ Si−2/tSi−2

(where we denote S−1 = M). For any element x ∈ Si we denote by x its residueclass in Si/tSi. Let f i(x) = 0, then fi(x) = ty for some y ∈ Si−1 and tfi−1(y) = 0.Since t is Si−2-regular, we have fi−1(y) = 0. Hence, there is x′ ∈ Si such thaty = fi(x

′). This implies that fi(x − tx′) = 0. Therefore, x− tx′ ∈ fi+1(Si+1) and

x ∈ f i+1(Si+1/tSi+1). Thus, the quotient sequence is exact.

The following proposition is often used as the definition of regular sequences:

Proposition A.3.3. A sequence t1, . . . , tk ∈ H(A+) is M -regular if and onlyif M is a free (not necessarily finitely generated) k[t1, . . . , tk]-module.

Proof. If M is a free k[t1, . . . , tk]-module, then M/(t1M + · · ·+ ti−1M) is afree k[ti, . . . , tk]-module, which implies that ti is M/(t1M + · · · + ti−1M)-regularfor 1 6 i 6 k. Therefore, t1, . . . , tk is an M -regular sequence.

Conversely, let t = (t1, . . . , tk) be an M -regular sequence. Consider a mini-mal resolution (Rmin, d) for the k[t ]-module M . Then, by Proposition A.3.2, thesequence of k-modules

· · · −→ R−1min/tR

−1min −→ R0

min/tR0min −→M/tM −→ 0

is exact. Note that R−imin/tR

−imin = R−i

min ⊗k[t ] k. Since the resolution is minimal,

the map R0min ⊗k[t ] k→M ⊗k[t] k is an isomorphism. Hence, R−i

min ⊗k[t] k = 0 for

i > 0, which implies that R−imin = 0. Thus, R0

min →M is an isomorphism, i.e. M isa free k[t ]-module.

The following is a direct corollary of Proposition A.3.3:

Proposition A.3.4. The property of being a regular sequence does not dependon the order of elements in t = (t1, . . . , tk).

Lemma A.3.5. Let t be a sequence of elements of A which is A-regular andM -regular. Then

Tor−iA (M,k) = Tor−iA/t (M/tM,k).

Proof. Applying Proposition A.3.2 to a minimal resolution of M , we obtaina minimal resolution of the A/t-module M/tM . The rest follows from Proposi-tion A.2.6.

An M -regular sequence is maximal if it is not contained in an M -regular se-quence of greater length.

Theorem A.3.6 (D. Rees). All maximal M -regular sequences in A have thesame length given by

(A.9) depthAM = mini : ExtiA(k,M) 6= 0

.

This number given by (A.9) is referred to as the depth of M ; the simplifiednotation depthM will be used whenever it creates no confusion. The proof ofTheorem A.3.6 uses the following fact:

Lemma A.3.7. Let t = (t1, . . . , tn) ∈ H(A+) be a M -regular sequence. Then

ExtnA(k,M) ∼= HomA(k,M/tM).


Proof. We use induction on n. The case n = 0 is tautological. It follows fromLemma A.3.4 that tn is an M -regular element, so we have the exact sequence

0 −→M·tn−→M −→M/tnM −→ 0.

The map ExtiA(k,M) → ExtiA(k,M) induced by multiplication by tn is zero (anexercise). Therefore, the second long exact sequence for Ext induced by the shortexact sequence above splits into short exact sequences of the form

0 −→ Extn−1A (k,M) −→ Extn−1

A (k,M/tnM) −→ ExtnA(k,M) −→ 0.

Let t ′ = (t1, . . . , tn−1). By induction,

Extn−1A (k,M) ∼= HomA(k,M/t ′M) = 0,

where the latter identity follows from Exercise A.3.17, since tn is M/t ′M -regular.Now the exact sequence above implies that

ExtnA(k,M) ∼= Extn−1A (k,M/tnM) ∼= HomA(k,M/tM),

where the latter identity follows by induction.

Proof of Theorem A.3.6. Let t = (t1, . . . , tn) be a maximal M -regular se-quence. Then, by Lemma A.3.7 and Exercise A.3.17,

ExtnA(k,M) ∼= HomA(k,M/tM) 6= 0,

as A does not contain an M/tM -regular element. On the other hand,

ExtiA(k,M) ∼= HomA

(k,M/(t1M + · · ·+ tiM)

)= 0

for i < n, since ti+1 is M/(t1M + · · ·+ tiM)-regular.

The following fundamental result relates the depth to the projective dimension:

Theorem A.3.8 (Auslander–Buchsbaum). Let M 6= 0 be an A-module suchthat pdimM <∞. Then

pdimM + depthM = depthA.

Proof. First let depthA = 0. Assume that pdimM = p > 0. Consider theminimal resolution for M (which is finite by hypothesis):

0→ R−pmin

dp−−−−→ R−p+1min −−−−→ · · · −−−−→ R0

min −−−−→ M → 0.

Since depthA = 0, we have HomA(k, A) = Ext0A(k, A) 6= 0 by Theorem A.3.6, i.e.there is a monomorphism of A-modules i : k→ A. In the commutative diagram

R−p ⊗A kdp⊗Ak−−−−→ R−p+1 ⊗A k

id⊗Ai

yyid⊗Ai

R−p dp−−−−→ R−p+1

the maps dp and id⊗A i are injective (the latter because the module R−p is free).Hence dp⊗A k is also injective, which contradicts minimality of the resolution. Weobtaine pdimM = 0, i.e. M is a free A-module and depthM = depthA = 0.

Now let depthA > 0. Assume that depthM = 0. Consider the first syzygymodule M1 = Ker[R0 → M ] for M . It follows from (A.9) and the exact sequencefor Ext that depthM1 = 1. Since pdimM1 = pdimM −1, it is enough to prove theAuslender–Buchsbaum formula for the module M1. Hence, we may assume that


depthM > 0. This implies that there is an element t ∈ A which is A-regular andM -regular (an exercise). Then

depthA/tA/t = depthAA− 1, depthA/tM/tM = depthAM − 1

by the definition of depth, and

pdimA/tM/tM = pdimAM

by Corollary A.2.7 and Lemma A.3.5. Now we finish by induction on depthA.

The dimension of A, denoted dimA, is the maximal number of elements of Aalgebraically independent over k. The dimension of an A-module M is dimM =dim(A/AnnM), where AnnM = a ∈ A : aM = 0 is the annihilator of M .

Definition A.3.9. A sequence t1, . . . , tn of algebraically independent homoge-neous elements of A is called a homogeneous system of parameters (briefly hsop) forM if dimM/(t1M+· · ·+tnM) = 0. Equivalently, t1, . . . , tn is an hsop if n = dimMand M is a finitely-generated k[t1, . . . , tn]-module.

An hsop consisting of linear elements (i.e. elements of lowest positive degree 2)is referred to as a linear system of parameters (briefly lsop).

The following result (due to Hilbert) is a graded version of the well-knownNoether Normalisation Lemma:

Theorem A.3.10 ([45, Theorem 1.5.17]). An hsop exists for any A-module M .If k is an infinite field and A is generated by degree-two elements, then a lsop canbe chosen for M .

It is easy to see that a regular sequence consists of algebraically independentelements, which implies that depthM 6 dimM .

Definition A.3.11. M is a Cohen–Macaulay A-module if depthM = dimM ,that is, if A contains an M -regular sequence t1, . . . , tn of length n = dimM . If Ais a Cohen–Macaulay A-module, then it is called a Cohen–Macaulay algebra.

By Proposition A.3.13, A is a Cohen–Macaulay algebra if and only if it is a freefinitely generated module over a polynomial subalgebra.

Proposition A.3.12. Let M be a Cohen–Macaulay A-module. Then a sequencet = (t1, . . . , tk) ∈ H(A+) is M -regular if and only if it is a part of an hsop for M .

Proof. Let dimM = n. Assume that t is an M -regular sequence. The factthat ti is an M/(t1M + · · ·+ ti−1M)-regular element implies that

dimM/(t1M + · · ·+ tiM) = dimM/(t1M + · · ·+ ti−1M)− 1, i = 1, . . . , k

(an exercise). Therefore, dimM/tM = n− k, i.e. t is a part of an hsop for M .For the other direction, see [45, Theorem 2.1.2 (c)].

In particular, any hsop in a Cohen–Macaulay algebra A is regular.

Proposition A.3.13. An algebra is Cohen–Macaulay if and only if it is a freefinitely generated module over a polynomial subalgebra.

Proof. Assume that A is Cohen–Macaulay and dimA = n. Then there is aregular sequence t = (t1, . . . , tn) in A. By the previous proposition, t is an hsop, sothat dimA/t = 0 and therefore A is a finitely generated k[t1, . . . , tn]-module. Thismodule is also free by Proposition A.3.3.


On the other hand, if A is free finitely generated over k[t1, . . . , tn] wheret1, . . . , tn ∈ A, then dimA = n and t1, . . . , tn is a regular sequence by Proposi-tion A.3.3, so that A is Cohen–Macaulay.

Proposition A.3.14. If A is Cohen–Macaulay with an lsop t = (t1, . . . , tn),then there is the following formula for the Poincare series of A:

F (A;λ) =F(A/(t1, . . . , tn);λ

)

(1− λ2)n ,

where F (A/(t1, . . . , tn);λ) is a polynomial with nonnegative integer coefficients.

Proof. Since A is a free finitely generated module over k[t1, . . . , tn], we havean isomorphism of k-vector spaces A ∼= (A/t) ⊗ k[t1, . . . , tn]. Calculating thePoincare series of both sides yields the required formula.

Remark. If A is generated by its elements a1, . . . , an of positive degreesd1, . . . , dn respectively, then it may be shown that the Poincare series of A is arational function of the form

F (A;λ) =P (λ)

(1 − λd1)(1 − λd2) · · · (1− λdn) ,

where P (λ) is a polynomial with integer coefficients. However, in general the poly-nomial P (λ) cannot be given explicitly, and some of its coefficients may be negative.

Exercises.

A.3.15. An M -regular sequence consists of algebraically independent elements.

A.3.16. The map ExtiA(k,M)→ ExtiA(k,M) induced by multiplication by anelement x ∈ H(A+) is zero.

A.3.17. The following conditions are equivalent for an A-module M :

(a) Every element of H(A+) is a zero divisor on M , i.e. depthM = 0;(b) HomA(k,M) 6= 0.

(Hint: show that if H(A+) consists of zero divisors on M then the ideal A+ anni-hilates a homogeneous element of M , see [106, Corollary 3.2].)

A.3.18. Let depthA > 0, let M be an A-module with depthM = 0, and letM1 = Ker[R0 →M ] be the first syzygy module for M . Then depthM1 = 1.

A.3.19. If depthA > 0 and depthM > 0, then there exists an element t ∈ Awhich is A-regular and M -regular.

A.3.20. Theorem A.3.8 does not hold if pdimM =∞.

A.3.21. Show that dimA = 0 if and only if A is finite-dimensional as a k-vectorspace. Is it true that depthA = 0 implies that dimkA is finite?

A.3.22. Give an example of an algebra A over a field k of finite characteristicwhich is generated by linear elements, but does not have an lsop.

A.3.23. A regular sequence consists of algebraically independent elements.

A.3.24. If t ∈ H(A+) is an M -regular element, then dimM/tM = dimM − 1.


A.3.25. Let k = Z. Show that if A is a free finitely generated module over apolynomial subalgebra Z[t1, . . . , tk] then ti+1 is not a zero divisor on A/(t1, . . . , ti)for 0 6 i < k, but the converse is not true. Therefore, the two possible definitionsof a regular sequence over Z do not agree. (The reason why Proposition A.3.13fails over Z is that minimal resolutions do not have the required good properties,see the remark after Construction A.2.2.)

A.4. Formality and Massey products

Here we develop the algebraic formalism used in rational homotopy theory. Wework with dg-algebras A =

⊕i>0A

i over a field k of zero characteristic (usually R

or Q). We do not assume A to be finitely generated. Commutativity of dg-algebrasis always understood in the graded sense.

A dg-algebra A is called homologically connected if H0(A, d) = k.Recall that a homomorphism between dg-algebras (A, dA) and (B, dB) is a

k-linear map f : A → B which preserves degrees, i.e. f(Ai) ⊂ Bi, and satisfiesf(ab) = f(a)f(b) and dBf(a) = f(dAa) for all a, b ∈ A. Such a homomorphism

induces a homomorphism f : H(A, dA) → H(B, dB) of cohomology algebras. We

refer to f as a quasi-isomorphism if f is an isomorphism. The equivalence relationgenerated by quasi-isomorphisms of dg-algebras is referred to as weak equivalence.Since quasi-isomorphisms are often not invertible, a weak equivalence between Aand B implies only the existence of a zigzag of quasi-isomorphisms of the form

A← A1 → A2 ← A3 → · · · ← Ak → B.

A dg-algebra B weakly equivalent to A is called a model of A. The above ‘long’zigzag of quasi-isomorphisms can be reduced to a ‘short’ zigzag A←M → B usingthe notion of a minimal model.

Definition A.4.1. A commutative dg-algebraM =⊕

i>0Mi is called minimal

(in the sense of Sullivan) if the following three conditions are satisfied:

(a) M0 = k and d(M0) = 0;(b) M is a free commutative dg-algebra, i.e.

M = Λ[xk : deg xk is odd]⊗ k[xk : deg xk is even],

there are only finitely many generators in each degree, and

deg xk 6 deg xl for k 6 l;

(c) dxk is a polynomial on generators x1, . . . , xk−1, for each k > 1 (this iscalled the nilpotence condition on d).

Clearly, a minimal dg-algebraM is simply connected (i.e. H1(M,d) = 0) if andonly if M1 = 0. In this case deg xk > 2 for each k, and the nilpotence condition isequivalent to the decomposability of d, i.e.

d(M) ⊂M+ ·M+,

where M+ is the subspace generated by elements of positive degree. For non-simply connected dg-algebras decomposability does not imply nilpotence: the alge-bra Λ[x, y], deg x = deg y = 1, with dx = 0, dy = xy is not minimal.

Definition A.4.2. A minimal dg-algebra M is called a minimal model for acommutative dg-algebra A if there is a quasi-isomorphism h : M → A.

A.4. FORMALITY AND MASSEY PRODUCTS 403

Theorem A.4.3.

(a) For each homologically connected commutative dg-algebra A satisfying thecondition dimHi[A] < ∞ for all i, there exists a minimal model MA,which is unique up to isomorphism.

(b) A homomorphism of commutative dg-algebras f : A→ B lifts to a homo-

morphism f : MA →MB closing the commutative diagram

MAf−−−−→ MB

hA

yyhB

Af−−−−→ B.

(c) If f is a quasi-isomorphism, then f is an isomorphism.

Remark. Minimal models can be also defined for dg-algebras A which do notsatisfy the finiteness condition dimHi[A] <∞, but we shall not need this.

This theorem is due to Sullivan (simply connected case) and Halperin (general).A proof can be found in [194, Theorem II.6] or [112, §12].

Corollary A.4.4. A weak equivalence between two commutative dg-algebrasA,B satisfying the condition of Theorem A.4.3 can be represented by a ‘short’ zigzagA←M → B of quasi-isomorphisms, where M is the minimal model for A (or B).

Definition A.4.5. A dg-algebra is A called formal if it is weakly equivalentto its cohomology H [A] (viewed as a dg-algebra with zero differential).

Corollary A.4.6. A commutative dg-algebra A with the minimal model MA

is formal if and only if MA in formal. In this case there is a zigzag of quasi-isomorphisms A←MA → H [A].

If A is formal with minimal model MA, then the minimal model can be recov-ered from the cohomology algebra H [A] using an inductive procedure.

Remark. Even if A is formal, the zigzag A←MA → H [A] usually cannot bereduced to a single quasi-isomorphism A→ H [A] or H [A]→ A.

Example A.4.7. Let M be a dg-algebra with three generators a1, a2, a3 ofdegree 1 and differential given by

da1 = da2 = 0, da3 = a1a2.

This M is minimal, but nor formal. Indeed the first degree cohomology H1[M ] isgenerated by the classes α1, α2 corresponding to the cocycles a1, a2, and we haveα1α2 = 0. Assume there is a quasi-isomorphism f : M → H [M ]; then we havef(a3) = k1α1 + k2α2 for some k1, k2 ∈ k. This implies that f(a1a3) = 0, which isimpossible since a1a3 represents a nontrivial cohomology class.

We next review Massey products, which provide a simple and effective tool forestablishing nonformality of a dg-algebra. Massey products constitute a series ofhigher-order operations (or brackets) in the cohomology of a dg-algebra, with thesecond-order operation coinciding with the cohomology multiplication, while thehigher-order brackets are only defined for certain tuples of cohomology classes. Weshall only consider triple (third-order) Massey products here.


Construction A.4.8 (triple Massey product). Let A be a dg-algebra, and letα1, α2, α3 be three cohomology classes such that α1α2 = α2α3 = 0 in H [A]. Choosetheir representing cocycles ai ∈ Aki , i = 1, 2, 3. Since the pairwise cohomologyproducts vanish, there are elements a12 ∈ Ak1+k2−1 and a23 ∈ Ak2+k3−1 such that

da12 = a1a2 and da23 = a2a3.

Then one easily checks that

(−1)k1+1a1a23 + a12a3

is a cocycle in Ak1+k2+k3−1. Its cohomology class is called a (triple) Massey productof α1, α2 and α3, and denoted by 〈α1, α2, α3〉.

More precisely, the Massey product 〈α1, α2, α3〉 is the set of all elements inHk1+k2+k3−1[A] obtained by the above procedure. Since there are choices of a12and a23 involved, the set 〈α1, α2, α3〉 may consist of more than one element. Infact, a12 is defined up to addition of a cocycle in Ak1+k2−1, and a23 is defined upto a cocycle in Ak2+k3−1. Therefore, any two elements in 〈α1, α2, α3〉 differ by anelement of the subset

α1 ·Hk2+k3−1[A] + α3 ·Hk1+k2−1[A] ⊂ Hk1+k2+k3−1[A],

which is called the indeterminacy of the Massey product 〈α1, α2, α3〉.A Massey product 〈α1, α2, α3〉 is called trivial (or vanishing) if it contains

zero. Clearly, a Massey product 〈α1, α2, α3〉 is trivial if and only if its image in thequotient algebra H [A]/(α1, α3) is zero.

Proposition A.4.9. Let f : A → B be a quasi-isomorphism of dg-algebras.Then all Massey products in H [A] are trivial if and only they are all trivial in H [B].

Proof. Assume that all Massey products in H [B] are trivial. Let 〈α1, α2, α3〉be a Massey product in H [A]. We define elements a1, a2, a3, a12, a23 ∈ A as inConstruction A.4.8, and set bi = f(ai), bij = f(aij). Let βi denote the cohomology

class corresponding to the cocycle bi. Since β1β2 = f(α1α2) = 0 and β2β3 = 0, theMassey product 〈β1, β2, β3〉 is defined. By the assumption, 0 ∈ 〈β1, β2, β3〉. Thismeans that we may choose b′12, b

′23 ∈ B in such a way that

b1b2 = db′12, b2b3 = db′23, and (−1)k1+1b1b′23 + b′12b3 = db123

for some b123 ∈ Bk1+k2+k3−2. Since db12 = b1b2, we have d(b′12 − b12) = 0. Sincef : A → B is a quasi-isomorphism, there is a cocycle c12 ∈ A such that f(c12) =b′12 − b12, and similarly there is a cocycle c23 ∈ A such that f(c23) = b′23 − b23. Set

a′12 = a12 + c12, a′23 = a23 + c23.

Then da′12 = da12 = a1a2 and similarly da′23 = a2a3. Therefore, the cohomologyclass of c = (−1)k1+1a1a

′23 + a′12a3 is a Massey product of α1, α2, α3. Then

f(c) = (−1)k1+1b1b′23 + b′12b3 = db123.

Since f is a quasi-isomorphism, c is a coboundary, i.e. c = da123 for some a123 ∈ A.Therefore, the Massey product 〈α1, α2, α3〉 is trivial.

The fact that the triviality of Massey products in H [A] implies their trivialityin H [B] is proved similarly (an exercise).

Corollary A.4.10. If a dg-algebra A is formal, then all Massey products inH [A] are trivial.

A.4. FORMALITY AND MASSEY PRODUCTS 405

Proof. Apply Proposition A.4.9 to a zigzag A ← · · · → H [A] of quasi-isomorphisms and use that fact that all Massey products for the dg-algebra H [A]with zero differential are trivial.

Exercises.

A.4.11. Let A → B be a quasi-isomorphism of dg-algebras. Show that thetriviality of Massey products in H [A] implies their triviality in H [B].

A.4.12. Find a nontrivial Massey product in the cohomology of the dg-algebraof Example A.4.7.

APPENDIX B

Algebraic topology

B.1. Homotopy and homology

Here we collect the basic constructions and facts, with almost no proofs. Forthe details the reader is referred to standard sources on algebraic topology, suchas [115], [151] or [248]. Unlike the other appendices, we do not include exerciseshere, as there would be too many of them.

All spaces here are topological spaces, and all maps are continuous. We denoteby k a commutative ring with unit (usually Z or a field; in fact an abelian group isenough until we start considering the multiplication in cohomology).

Basic homotopy theory. Two maps f, g : X → Y of spaces are homotopic(denoted by f ≃ g) if there is a map F : X × I → Y (where I = [0, 1] is the unitinterval) such that F |X×0 = f and F |X×1 = g. We denote the map F |X× t : X → Yby Ft, for t ∈ I. Homotopy is an equivalence relation, and we denote by [X,Y ] theset of homotopy classes of maps from X to Y .

Two spaces X and Y are homotopy equivalent if there are maps f : X → Y andg : Y → X such that g f and f g are homotopic to the identity maps of X andY respectively. The homotopy type of a space X is the class of spaces homotopyequivalent to X .

A space X is contractible if it is homotopy equivalent to a point.A pair (X,A) of spaces consists of a space X and its subspace A. A map of

pairs f : (X,A)→ (Y,B) is a continuous map f : X → Y such that f(A) ⊂ B.A pointed space (or based space) is a pair (X, pt) where pt is a point of X , called

the basepoint. We denote by X+ the pointed space (X ⊔ pt , pt), where X ⊔ pt is Xwith a disjoint point added. A map of pointed spaces (or pointed map) is a map ofpairs (X, pt)→ (Y, pt). We denote a pointed space (X, pt) simply by X wheneverthe choice of the basepoint is clear or irrelevant. Given two pointed spaces (X, pt)and (Y, pt), their wedge (or bouquet) is defined as the pointed space X∨Y obtainedby attaching X and Y at the basepoints. Then X ∨ Y is contained as a pointedsubspace in the product X × Y , and the quotient X ∧ Y = (X × Y )/(X ∨ Y ) iscalled the smash product of X and Y .

For any pointed space (X, pt) the set of homotopy classes of pointed maps(Sk, pt)→ (X, pt) (where Sk is a k-dimensional sphere, k > 0) is a group for k > 0,which is called the kth homotopy group of (X, pt) and denoted by πk(X, pt) or sim-ply πk(X). We have that π0(X) is the set of path connected components of X . Thegroup π1(X) is called the fundamental group of X . The groups πk(X) are abelianfor k > 1. A pointed map f : X → Y induces a homomorphism f∗ : πk(X)→ πk(Y )for each k, and the homomorphisms induced by homotopic maps are the same.

407

408 B. ALGEBRAIC TOPOLOGY

A locally trivial fibration (or a fibre bundle) is a quadruple (E,B, F, p) whereE, B, F are spaces and p is a map E → B such that for any point x ∈ B there is

a neighbourhood U ⊂ B and a homeomorphism ϕ : p−1(U)∼=−→ U × F closing the

commutative diagram

p−1(U)ϕ //

p##

U × F

||②②②②②②②②②

B

The space E is called the total space, B the base, and F the fibre of the fibre bundle.The terms ‘locally trivial fibration’ and ‘fibre bundle’ are also often used for themap p : E → B.

A cell complex (or a CW-complex ) is a Hausdorff topological space X repre-sented as a union

⋃eqi of pairwise nonintersecting subsets eqi , called cells, in such a

way that for each cell eqi there is a map of a closed q-disk Dq to X (the characteris-tic map of eqi ) whose restriction to the interior of Dq is a homeomorphism onto eqi .Furthermore, the following two conditions are assumed:

(C) the boundary eqi \ eqi of a cell is contained in a union of finitely many cellsof dimensions < q;

(W) a subset Y ⊂ X is closed if and only the intersection Y ∩ eqi is closed forevery cell eqi (i.e. the topology of X is the weakest topology in which allcharacteristic maps are continuous).

The union of cells of X of dimension 6 n is called the nth skeleton of X and denotedby sknX or by Xn. A cell complex X can be obtained from its 0th skeleton sk0X(which is a discrete set) by iterating the operation of attaching a cell : a space Zis obtained from Y by attaching an n-cell along a map f : Sn−1 → Y if Z is thepushout of the form

(B.1)

Sn−1 f−−−−→ Yy

y

Dn −−−−→ Z

We use the notation Z = Y ∪f Dn.A cell subcomplex of a cell complex X is a closed subspace which is a union of

cells of X . Each skeleton of X is a cell subcomplex.A map f : X → Y between cell complexes is called a cellular map if f(sknX) ⊂

sknY for all n.

Theorem B.1.1 (Cellular approximation). A map between cell complexes ishomotopic to a cellular map.

For any integer n > 0 and any group π (which is assumed to be abelian ifn > 1) there exists a connected cell complex K(π, n) such that πn

(K(π, n)

) ∼= π

and πk(K(π, n)

)= 0 for k 6= n. (A cell complex K(π, n) can be constructed by

taking the wedge of n-spheres corresponding to a set of generators of π and thenkilling the higher homotopy groups by attaching cells.) The space K(π, n) is calledthe Eilenberg–Mac Lane space (corresponding to n and π), and it is unique up tohomotopy equivalence. Examples of Eilenberg–Mac Lane spaces include the circle

B.1. HOMOTOPY AND HOMOLOGY 409

S1 = K(Z, 1), the infinite-dimensional real projective space RP∞ = K(Z2, 1) andthe infinite-dimensional complex projective space CP∞ = K(Z, 2).

A space X which is homotopy equivalent to K(π, 1) for some π is called as-pherical . Any surface (a closed 2-dimensional manifold) which not a 2-sphere or areal projective plane is aspherical.

A locally trivial fibration p : E → B satisfies the following covering homotopyproperty (CHP for short) with respect to maps of cell complexes Y : for any homo-topy F : Y × I→ B and any map f : Y → E such that pf = F0 there is a covering

homotopy F : Y × I → E, satisfying F0 = f and p F = F . This is described bythe commutative diagram

(B.2) Yf //

i0

E

p

Y × I

F //

F

<<①①

①①

B.

A map p : E → B satisfying the CHP with respect to maps of cell complexesY → B is called a Serre fibration. A map p : E → B satisfying CHP with respectto all maps Y → B is called a Hurewitz fibration. The difference between Serreand Hurewitz fibrations is not important for our constructions and will be ignored;we shall use the term fibration for both of them. A fibre bundle is a fibration, butnot every fibration is a fibre bundle. All fibres p−1(b), b ∈ B, of a fibration arehomotopy equivalent if B is connected.

Theorem B.1.2 (Exact sequence of fibration). For a fibration p : E → B withfibre F there exists a long exact sequence

· · · → πk(F )i∗−→ πk(E)

p∗−→ πk(B)∂−→ πk−1(F )→ · · ·

where the map i∗ is induced by the inclusion of the fibre i : F → E, the map p∗ isinduced by the projection p : E → B, and ∂ is the connecting homomorphism.

The connecting homomorphism ∂ is defined as follows. Take an elementγ ∈ πk(B) and choose a representative g : Sk → B in the homotopy class γ. By

considering the composition Sk−1 × I → Skg−→ B (where the first map contracts

the top and bottom bases of the cylinder to the north and south poles of thesphere), we may view the map g as a homotopy F : Sk−1×I→ B of the trivial mapF0 : S

k−1 → pt with F1 also being trivial. Using the CHP we lift F to a homotopy

F : Sk−1 × I → E with F0 still trivial, but F1 : Sk−1 → E being trivial only after

projecting onto B. The latter condition means that F1 is in fact a map Sk−1 → F ,homotopy class of which we take for ∂γ.

The path space of a space X is the space PX of pointed maps (I, 0)→ (X, pt).The loop space ΩX is the space of pointed maps f : (I, 0) → (X, pt) such thatf(1) = pt . The map p : PX → X given by p(f) = f(1) is a fibration, with fibre(homotopy equivalent to) ΩX .

Proposition B.1.3. For any map f : X → Y there exist a homotopy equiva-

lence h : X → X and a fibration p : X → Y such that f = p h. Furthermore, this


decomposition is functorial in the sense that a commutative diagram of maps

X //

f

X ′

f ′

Y // Y ′

induces a commutative diagram

X //

h

f

X ′

f ′

h′

X

p⑧⑧⑧⑧⑧⑧⑧⑧

// X ′

p′~~⑥⑥⑥⑥⑥⑥⑥⑥

Y // Y ′

Proof. Let X be the set of pairs (x, g) consisting of a point x ∈ X and a pathg : I→ Y with g(0) = f(x).This is described by the pullback diagram

X //

Y I

p0

X

f // Y

where Y I is the space of all paths g : I→ Y and the map p0 takes g to g(0).

Then the homotopy equivalence h : X → X is given by h(x) = (x, cf(x)), where

cf(x) : I → Y is the constant path t 7→ f(x), and the fibration p : X → Y is givenby p(x, h) = h(1). The functoriality property follows by inspection.

A space homotopy equivalent to the fibre of the fibration p : X → Y fromProposition B.1.3 is referred to as the homotopy fibre of the map f : X → Y , and

denoted by hofib f . The functoriality of the construction of X implies that thehomotopy fibre is well-defined: for any other decomposition f = p′ h′ into acomposition of a fibration p′ and a homotopy equivalence h′ the homotopy fibre off is homotopy equivalent to the fibre of p′. The homotopy fibre of the inclusionpt → X is the loop space ΩX .

An inclusion i : A → X of a cell subcomplex in a cell complex X satisfies thefollowing homotopy extension property (HEP for short): for any map f : X → Y ,a homotopy F : A × I → Y such that F0 = f |A can be extended to a homotopy

F : X × I → Y such that F0 = f and F |A×I = F . This is described by thecommutative diagram

(B.3) AF ′

//

i

Y I

p0

X

f //

F ′

>>⑥⑥

⑥⑥

Y

where F ′ is the adjoint of F (i.e. F ′(a) = h where h(t) = F (a, t) ∈ Y ), the map p0takes h to h(0), and F ′ is the adjoint of F .


A map i : A → X satisfying the HEP is called a cofibration, and X/i(A) is itscofibre. A pair (X,A) for which i : A→ X is a cofibration is called a Borsuk pair.

Diagram (B.2) expresses the fact that fibrations obey the right lifting prop-erty with respect to particular cofibrations Y → Y × I, which are also homotopyequivalences. Similarly, diagram (B.3) expresses the fact that cofibrations obey theleft lifting property with respect to particular fibrations Y I → Y , which are alsohomotopy equivalences. This will be important for axiomatising homotopy theoryvia the concept of model category, see Section C.1.

The cone over a space X is the quotient space coneX = (X × I)/(X × 1).The suspension ΣX is the quotient (coneX)/(X × 0). There are inclusions X →coneX → ΣX of closed subspaces, where the first map is given by x 7→ (x, 0) andthe second by (x, t) 7→

(x, (t+1)/2

)for x ∈ X , t ∈ I. The inclusion i : X → coneX

is a cofibration, with cofibre ΣX .

Proposition B.1.4. For any map f : X→ Y there exist a cofibration i : X→ Y

and a homotopy equivalence h : Y → Y such that f = h i. This decomposition isfunctorial.

Proof. Let Y be the quotient of (X×I)⊔Y obtained by identifying x×0 ∈ X×Iwith f(x) ∈ Y . This is described by the pushout diagram

Xf //

i0

Y

X × I // Y

where the map i0 takes x to x× 0.

The cofibration i : X → Y is given by i(x) = x× 1, and the homotopy equiva-

lence h : Y → Y is given by h(x× t) = f(x) and h(y) = y.

The space Y from Proposition B.1.4 is known as the mapping cylinder of the

map f : X → Y . The space Y /i(X) = Y /(X×1) is known as the mapping cone off : X → Y . A space homotopy equivalent to the mapping cone of f is called thehomotopy cofibre of the map f : X → Y . The homotopy cofibre of the projectionX → pt is the suspension ΣX .

Simplicial homology and cohomology. Let K be a simplicial complexon the set [m] = 1, . . . ,m. An oriented q-simplex σ of K is a k-simplexI = i1, . . . , iq+1 ∈ K together with an equivalence class of total orderings ofthe set I, two orderings being equivalent if they differ by an even permutation. De-note by [I] the oriented q-simplex with the equivalence class of orderings containingi1 < · · · < iq+1.

Define the qth simplicial chain group (or module) Cq(K;k) with coefficientsin k as the free k-module with basis consisting of oriented q-simplices of K, modulothe relations σ + σ = 0 whenever σ and σ are differently oriented q-simplicescorresponding to the same simplex of K. Therefore, Cq(K,k) is a free k-module ofrank fq(K) (the number of q-simplices of K) for q > 0, and Cq(K;k) = 0 for q < 0.


Define the simplicial chain boundary homomorphisms

∂q : Cq(K;k)→ Cq−1(K;k), q > 1,

∂q[i1, . . . , iq+1] =

q+1∑

j=1

(−1)j−1[i1, . . . , ij, . . . , iq+1],

where ij denotes that ij is missing. It is easily checked that ∂q∂q+1 = 0, so thatC∗(K;k) = Cq(K;k), ∂q is a chain complex, called the simplicial chain complexof K (with coefficients in k).

The qth simplicial homology group of K with coefficients in k, denoted byHq(K;k), is defined as the qth homology group of the simplicial chain complexC∗(K;k).

The Euler characteristic χ(K) of K is defined by

χ(K) =∑

q>0

(−1)q rankHq(K;k).

It is also given by

χ(K) = f0 − f1 + f2 − · · · ,where fi is the number of i-simplices of K, and therefore χ(K) is independent of k.

The augmented simplicial chain complex C∗(K;k) is obtained by taking into

account the empty simplex ∅ ∈ K. That is, Cq(K;k) = Cq(K;k) for q 6= −1 and

C−1(K;k) ∼= k is a free k-module with basis [∅], so that C∗(K;k) is written as

· · · → Cq(K;k)∂q−→ · · · −→ C1(K;k) ∂1−→ C0(K;k) ε−→ C−1(K;k)→ 0,

where ε = ∂0 is the augmentation taking each vertex [i] to [∅].

The qth homology group of C∗(K;k) is called the qth reduced simplicial homol-

ogy group of K with coefficients in k and is denoted by Hq(K;k). We have that

Hq(K;k) = Hq(K;k) for q > 1, and H0(K;k) ∼= H0(K;k) ⊕ k unless K consists of

∅ only, in which case H−1(∅;k) ∼= k.The simplicial cochain complex of K with coefficients in k is defined to be

C∗(K;k) = Homk

(C∗(K;k),k

).

In explicit terms, C∗(K;k) = Cq(K;k), dq. Here Cq(K;k) is the qth simplicialcochain group (or module) of K with coefficients in k; it is a free k-module withbasis consisting of simplicial cochains αI corresponding to q-simplices I ∈ K; thecochain αI takes value 1 on the oriented simplex [I] and vanishes on all otheroriented simplices. The value of the cochain differential

dq = ∂∗q+1 : Cq(K;k)→ Cq+1(K;k)

on the basis elements is given by

dαI =∑

j∈[m]\I, j∪I∈K

ε(j, j ∪ I)αj∪I ,

where the sign is given by ε(j, j ∪ I) = (−1)r−1 if j is the rth element of the setj ∪ I ⊂ [m], written in increasing order.

The qth simplicial cohomology group of K with coefficients in k, denoted byHq(K;k), is defined as the qth cohomology group of the cochain complex C∗(K;k).


The qth reduced simplicial cohomology group of K with coefficients in k, denoted

by Hq(K;k), is defined as the qth cohomology group of the cochain complex

0→ kd−1−→ C0(K;k) d0−→ C1(K;k) d1−→ · · · −→ Cq(K;k) dq−→ · · ·

obtained by applying the functor Hom to the augmented chain complex C∗(K;k).The map d−1 takes 1 = α∅ to the sum of αi corresponding to all vertices i ∈ K.

We have that Hq(K;k) = Hq(K;k) for q > 1, and H0(K;k) ∼= H0(K;k)⊕k unless

K consists of ∅ only, in which case H−1(∅;k) ∼= k.

Singular homology and cohomology. Let X be a topological space. Asingular q-simplex of X is a continuous map f : ∆q → X . The qth singular chaingroup Cq(X ;k) of X with coefficients in k is the free k-module generated by allsingular q-simplices.

For each i = 1, . . . , q + 1 there is a linear map ϕiq : ∆q−1 → ∆q which sends

∆q−1 to the face of ∆q opposite its ith vertex, preserving the order of vertices.The ith face of a singular simplex f : ∆q → X , denoted by f (i) is defined to be thesingular (q − 1)-simplex given by the composition

f (i) = f ϕiq : ∆q−1 → ∆q → X.

The singular chain boundary homomorphisms are given by

∂q : Cq(X ;k)→ Cq−1(X ;k), q > 1,

∂qf =

q+1∑

i=1

(−1)i−1f (i).

Then ∂q∂q+1 = 0, so that C∗(X ;k) = Cq(X ;k), ∂q is a chain complex, called thesingular chain complex of X (with coefficients in k).

The qth singular homology group of X with coefficients in k, denoted byHq(X ;k), is the qth homology group of the singular chain complex C∗(X ;k).

Assume that a space X has only finite number of nontrivial homology groups(with Z coefficients), and each of these groups has finite rank. Then the Eulercharacteristic χ(X) of X is defined by

χ(X) =∑

q>0

(−1)q rankHq(X ;Z).

Proposition B.1.5. The group H0(X ;k) is a free k-module of rank equal tothe number of path connected components of X.

The augmented singular chain complex C∗(X ;k) is defined by Cq(X ;k) =

Cq(X ;k) for q 6= −1 and C−1(X ;k) = k; the augmentation ε = ∂0 : C0(X ;k)→ k

is given by ε(f) = 1 for all singular 0-simplices f .

The qth homology group of C∗(X ;k) is called the qth reduced singular homology

group of K with coefficients in k and is denoted by Hq(X ;k). We have Hq(X ;k) =

Hq(X ;k) for q > 1, and H0(X ;k) ∼= H0(X ;k)⊕ k if X is nonempty.

A nonempty space X is acyclic if Hq(X ;Z) = 0 for all q.We shall drop the coefficient group k in the notation of chains and homology

occasionally.


For any simplicial complex K, there is an obvious inclusion i : Cq(K)→ Cq(|K|)of the simplicial chain groups into the singular chain groups of the geometric reali-sation K, defining an inclusion of chain complexes.

Theorem B.1.6. The map i : Cq(K) → Cq(|K|) induces an isomorphism

Hq(K)∼=−→ Hq(|K|) between the simplicial homology groups of K and the singu-

lar homology groups of |K|.If (X,A) is pair of spaces, then C∗(A) is a chain subcomplex in C∗(X) (that is,

Cq(A) is a submodule of Cq(X) and ∂qCq(A) ⊂ Cq−1(A)). The quotient complex

C∗(X,A) =Cq(X,A) = Cq(X)/Cq(A), ∂q : Cq(X,A)→ Cq−1(X,A)

is called the singular chain complex of the pair (X,A). Its qth homology groupHq(X,A) is called the qth singular homology group of (X,A), or qth relative singular

homology group of X modulo A. Note that Hq(X) = Hq(X,∅) and Hq(X) =Hq(X, pt).

The singular cochain complex C∗(X ;k) = Cq(X ;k), dq is defined to be

C∗(X ;k) = Homk

(C∗(X ;k),k

).

Its qth cohomology group is called the qth singular cohomology group of X and

denoted by Hq(X ;k). The reduced singular cohomology groups Hq(X ;k) and therelative cohomology groups Hq(X,A) are defined similarly.

The ranks of the groups Hq(X ;Z) and Hq(X ;Z) coincide, and the numberbq(X) = rankHq(X ;Z) is called the qth Betti number of X . The Betti numberswith coefficients in a field k are defined similarly.

The (co)homology groups have the following fundamental properties.

Theorem B.1.7 (Functoriality and homotopy invariance). A map f : X → Yinduces homomorphisms f∗ : Hq(X ;k)→ Hq(Y ;k) and f∗ : Hq(Y ;k)→ Hq(X ;k).If two maps f, g : X → Y are homotopic, then f∗ = g∗ and f∗ = g∗.

We omit the coefficient group k in the notation for the rest of this subsection.

Theorem B.1.8 (Exact sequences of pairs). For any pair (X,A) there are longexact sequences

· · · → Hq(A)i∗−→ Hq(X)

j∗−→ Hq(X,A)∂−→ Hq−1(A)→ · · · ,

· · · → Hq(X,A)j∗−→ Hq(X)

i∗−→ Hq(A)d−→ Hq+1(X,A)→ · · ·

Furthermore, if the inclusion A → X is a cofibration (e.g., if it is an inclusionof a cell subcomplex), then the quotient projection (X,A) → (X/A, pt) inducesisomorphisms

Hq(X,A) ∼= Hq(X/A, pt) = Hq(X/A), Hq(X,A) ∼= Hq(X/A, pt) = Hq(X/A).

In the homology exact sequence of a pair, the map i∗ is induced by the inclusionA → X , the map j∗ is induced by the map of pairs (X,∅) → (X,A), and theconnecting homomorphism ∂ is the homology homomorphism corresponding theboundary homomorphism sending a relative cocycle c ∈ Cq(X,A) to ∂c ∈ Cq(A).The maps in the cohomology exact sequence are defined dually.

Let (X,A) be a pair and B ⊂ A be a subspace. The inclusion of pairs(X\B,A\B)→ (X,A) induces maps

Hq(X\B,A\B)→ Hq(X,A),


which are referred to as the excision homomorphisms.

Theorem B.1.9 (Excision). If the closure of B is contained in the interior of A,then the excision homomorphisms Hq(X\B,A\B)→ Hq(X,A) are isomorphisms.

Theorem B.1.10 (Mayer–Vietoris exact sequences). Let X be a space, A ⊂ X,B ⊂ X and A ∪B = X. Assume that the excision homomorphisms

Hq(B,A ∩B)→ Hq(X,A) and Hq(A,A ∩B)→ Hq(X,B)

are isomorphisms for all q. Then there are exact sequences

· · · → Hq(A ∩B)β−→ Hq(A) ⊕Hq(B)

α−→ Hq(X)δ−→ Hq−1(A ∩B)→ · · · ,

· · · → Hq(X)α−→ Hq(A) ⊕Hq(B)

β−→ Hq(A ∩B)δ−→ Hq+1(X)→ · · · .

In the homology Mayer–Vietoris sequence, the map β is the difference of thehomomorphism induced by the inclusions A ∩B → A and A ∩B → B, the map αis the sum of the homomorphism induced by the inclusions A → X and B → X ,and the connecting homomorphism δ is the composite

Hq(X)j∗−→ Hq(X,A) = Hq(B,A ∩B)

∂−→ Hq−1(A ∩B).

The first key calculation of homology groups follows directly from the generalproperties above:

Proposition B.1.11. The n-disk Dn is acyclic, and the reduced homology of

an n-sphere Sn is given by Hn(Sn;k) ∼= k and Hi(S

n;k) = 0 for i 6= n.

Here is another direct corollary:

Theorem B.1.12 (Suspension isomorphism). For any space X and any q > 0there are isomorphisms

Hq(ΣX) ∼= Hq−1(X) and Hq(ΣX) ∼= Hq−1(X).

A closed connected topological n-dimensional manifold X is orientable over k

if Hn(X ;k) ∼= k. Every compact connected manifold is oriented over Z2 or anyfield of characteristic two.

Theorem B.1.13 (Poincare duality). If a closed connected n-manifold is ori-entable over k, then Hq(X ;k) ∼= Hn−q(X ;k).

Cellular homology, and cohomology multiplication. Let X be a cellcomplex with skeletons sknX = Xn for n = 0, 1, 2, . . .

We start with the case k = Z, and omit this coefficient group in the notation.The group Cq(X) = Hq(X

q, Xq−1) is called the qth cellular chain group. This is afree abelian group with basis the q-dimensional cells of X , and therefore a cellularchain may be viewed as an integral combination of cells of X .

The cellular boundary homomorphism ∂q : Cq(X) → Cq−1(X) is defined as thecomposition

∂q : Cq(X) = Hq(Xq, Xq−1)→ Hq−1(X

q−1)→ Hq−1(Xq−1, Xq−2) = Cq−1(X)

of homomorphisms from the homology exact sequences of pairs (Xq, Xq−1) and(Xq−1, Xq−2). The resulting chain complex

· · · → Cq(X)∂q−→ · · · −→ C1(X)

∂1−→ C0(X)→ 0


is called the cellular chain complex of X . Its qth homology group, which we denoteby Hq(X) for the moment, is called the qth cellular homology group of X .

Theorem B.1.14. For any cell complex X, there is a canonical isomorphismHq(X) ∼= Hq(X) between the cellular and singular homology groups.

The cellular homology groups Hq(X ;k) with coefficients in k and the cohomol-ogy groups Hq(X ;k) are defined similarly; they are canonically isomorphic to theappropriate singular homology and cohomology groups.

We shall therefore not distinguish between the singular and cellular homologyand cohomology groups of cell complexes.

The product X1 ×X2 of cell complexes is a cell complex with cells of the forme1 × e2, where e1 is a cell of X1 and e2 is a cell of X2.

Given two cellular cochains c1 ∈ Cq1(X) and c2 ∈ Cq2(X), define their productas the cochain c1 × c2 ∈ Cq1+q2(X ×X) whose value on a cell e1 × e2 of X ×X isgiven by (−1)q1q2c1(e1)c2(e2). This product satisfies the identity

δ(c1 × c2) = δc1 × c2 + (−1)q1c1 × δc2,where δ is the cellular cochain differential, and therefore defines a map of cochaincomplexes

C∗(X)⊗ C∗(X)→ C∗(X ×X)

and induces a cohomology map

× : Hq1(X)⊗Hq2(X)→ Hq1+q2(X ×X),

which is called the (cohomology) ×-product.The composition of the ×-product with the cohomology map induced by the

diagonal map ∆ : X → X ×X , ∆(x) = (x, x), defines a product

` : Hq1(X)⊗Hq2(X)×−→ Hq1+q2(X ×X)

∆∗

−→ Hq1+q2(X),

which is called the cup product, or cohomology product. It turns H∗(X) =⊕q>0H

q(X) into an associative and graded commutative ring. This ring structure

on H∗(X) is a homotopy invariant of X ; it does not depend on the cell complexstructure. It is also functorial, in the sense that a map f : X → Y induces a ringhomomorphism f∗ : H∗(Y )→ H∗(X).

There is also a relative version of the cohomology product, given by the map

Hq1(X ;A)⊗Hq2(X,B) −→ Hq1+q2(X,A ∪B).

Whitehead product, Samelson product and Pontryagin product. Letw : Sk+l−1 → Sk ∨ Sl be the attaching map of the (k + l)-cell of Sk × Sl with thestandard cell structure. Explicitly, the map w can be defined as the composition

Sk+l−1 = ∂(Dk ×Dl) = Dk × Sl−1 ∪Sk−1×Sl−1 Sk−1 ×Dl → Sk ∨ Sl,where the last map consists of two projections

Dk × Sl−1 → Dk → Dk/Sk−1 = Sk → Sk ∨ Sl and

Sk−1 ×Dl → Dl → Dl/Sl−1 = Sl → Sk ∨ Sl

and maps Sk−1 × Sl−1 to the basepoint.


Given two pointed maps f : Sk → X and g : Sl → X , their Whitehead productis defined as the composition

[f, g]w : Sk+l−1 w−→ Sk ∨ Sl f∨g−→ X.

It gives rise to a well-defined product

[ · , · ]w : πk(X)× πl(X)→ πk+l−1(X),

which is also called the Whitehead product. When k = l = 1, the Whiteheadproduct is the commutator product in π1(X). We have [f, g]w = 0 in πk+l−1(X)whenever the map f ∨ g : Sk ∨ Sl → X extends to a map Sk × Sl → X .

Theorem B.1.15.

(a) If α ∈ πk(X) and β, γ ∈ πl(X) with l > 1, then

[α, β + γ]w = [α, β]w + [α, γ]w.

(b) If α ∈ πk(X) and β ∈ πl(X) with k, l > 1, then

[α, β]w = (−1)kl[β, α]w .(c) If α ∈ πk(X), β ∈ πl(X) and γ ∈ πm(X) with k, l,m > 1, then

(−1)km[[α, β]w , γ]w + (−1)lk[[β, γ]w, α]w + (−1)ml[[γ, α]w, β]w = 0.

Now consider the loop space ΩX . The commutator of loops, (x, y) 7→xyx−1y−1, induces a map c : ΩX ∧ ΩX → ΩX . Given two pointed mapsf : Sp → ΩX and g : Sq → ΩX , their Samelson product is defined as

[f, g]s : Sp+q = Sp ∧ Sq f∧g−→ ΩX ∧ΩX c−→ ΩX.

It gives rise to a well-defined product

[ · , · ]s : πp(ΩX)× πq(ΩX)→ πp+q(ΩX),

which is also called the Samelson product.

Theorem B.1.16.

(a) If ϕ ∈ πp(ΩX) and ψ, η ∈ πq(ΩX), then

[ϕ, ψ + η]s = [ϕ, ψ]s + [ϕ, η]s.

(b) If ϕ ∈ πp(ΩX) and ψ ∈ πq(ΩX), then

[ϕ, ψ]s = −(−1)pq[ψ, ϕ]s.(c) If ϕ ∈ πp(ΩX), ψ ∈ πq(ΩX) and η ∈ πr(ΩX), then

[ϕ, [ψ, η]s]s = [[ϕ, ψ]s, η]s + (−1)pq[ψ, [ϕ, η]s]s.The Samelson bracket makes the rational vector space π∗(ΩX) ⊗ Q into a

graded Lie algebra, see (C.8). It is called the rational homotopy Lie algebra of X .The Pontryagin product is defined as the composition

H∗(ΩX ;k)⊗H∗(ΩX ;k)×−→ H∗(ΩX ×ΩX ;k)

m∗−→ H∗(ΩX ;k),

where k is a commutative ring with unit, × is the homology cross-product, andm : ΩX × ΩX → ΩX is the loop multiplication. Pontryagin product makesH∗(ΩX ;k) into an associative (but not generally commutative) algebra with unit.

Whitehead, Samelson and Pontryagin products are related by the followingclassical result of Samelson:


Theorem B.1.17 ([280]). There is a choice of adjunction isomorphismt : πn(X)→ πn−1(ΩX) such that

t[α, β]w = (−1)k−1[tα, tβ]s

for α ∈ πk(X) and β ∈ πl(X). Furthermore, if h : πn(ΩX) → Hn(ΩX) is theHurewicz homomorphism and ⋆ denotes the Pontryagin product, then

h[ϕ, ψ]s = h(ϕ) ⋆ h(ψ)− (−1)pqh(ψ) ⋆ h(ϕ)for ϕ ∈ πp(ΩX) and ψ ∈ πq(ΩX).

It follows that the Pontryagin algebra H∗(ΩX,Q) is the universal envelopingalgebra of the graded Lie algebra π∗(ΩX)⊗Q, and π∗(ΩX)⊗Q is the Lie algebraof primitive elements in the Hopf algebra H∗(ΩX,Q).

More details on the relationship between the three products, including theirgeneralisations to H-spaces, can be found in the monograph by Neisendorfer [240].

B.2. Elements of rational homotopy theory

We only review some basis notions and results here. For a detailed account ofthis much elaborated theory we refer to [39], [194] and [112].

Definition B.2.1. A map f : X → Y between spaces is called a rationalequivalence if it induces isomorphisms in all rational homotopy groups, that is,f∗ : πi(X)⊗Q→ πi(Y )⊗Q is an isomorphism for all i.

The rational homotopy type of X is its equivalence class in the equivalencerelation generated by rational equivalences.

Proposition B.2.2. If both X and Y are simply connected, then f : X → Yis a rational equivalence if and only if f∗ : Hi(X ;Q) → Hi(Y ;Q) (or equivalently,f∗ : Hi(Y ;Q)→ Hi(X ;Q)) is an isomorphism for each i.

A space X is nilpotent if its fundamental group π1(X) is nilpotent and π1(X)acts nilpotently on higher homotopy groups πn(X). Simply connected spaces areobviously nilpotent.

Rational homotopy theory, whose foundation was laid in the work of Quillen[271] and Sullivan [298], translates the study of the rational homotopy type ofnilpotent spaces X into the algebraic language of dg-algebras and minimal mod-els (see Section A.4). This translation is made via Sullivan’s algebra of piecewisepolynomial differential forms APL(X), whose properties we briefly discuss below.Further remarks relating rational homotopy theory to the theory of model categoriesare given in Section C.1.

The most basic dg-algebra model of a spaceX is its singular cochains C∗(X ;k).However, this dg-algebra is non-commutative, and therefore is difficult to handle.If X is a smooth manifold, and k = R, then we may consider the dg-algebraΩ∗(X) of de Rham differential forms instead of C∗(X ;R). It provides a functorial(with respect to smooth maps) and commutative dg-algebra model for X , withthe same cohomology H∗(X ;R). It is therefore natural to ask whether a functorialcommutative dg-algebra model exists for arbitrary cell complexesX over a field k ofcharacteristic zero (over finite fields there are secondary cohomological operationsobstructing such a construction). A first construction of a commutative dg-algebramodel which worked for simplicial complexes was suggested by Thom in the end ofthe 1950s. Later, in the mid-1970s, Sullivan provided a functorial construction of

B.2. ELEMENTS OF RATIONAL HOMOTOPY THEORY 419

a dg-algebra model APL(X) whose cohomology is H∗(X ;Q), using similar ideas asthose of Thom (a combinatorial version of differential forms).

The algebra APL(X) has the following two important properties.

Theorem B.2.3 (PL de Rham Theorem).

(a) APL(X) is weakly equivalent to C∗(X ;Q) via a short zigzag of the form

APL(X)→ D(X)← C∗(X ;Q),

where D(X) is another naturally defined dg-algebra;(b) there is a natural map of cochain complexes APL(X) → C∗(X ;Q), the

‘Stokes map’, which induces an isomorphism in cohomology.(c) if X is a smooth manifold, then the dg-algebra Ω∗(X) of de Rham forms

is weakly equivalent to APL(X)⊗Q R.

The proof of (a) can be found in [194, §III.3] or in [112, Corollary 10.10].For (b), see [39, §2], and for (c), see [194, Theorem V.2] or [112, Theorem 11.4].

Definition B.2.4. Given a cell complex X , we refer to any commutative dg-algebra A weakly equivalent to APL(X) as a (rational) model of X . The minimalmodel of APL(X) is called the minimal model of X , and is denoted by MX .

In the case of nilpotent spaces, the rational homotopy type of X is fully de-termined by the commutative dg-algebra APL(X) or its minimal model MX . Moreprecisely, there is the following fundamental result.

Theorem B.2.5. There is a bijective correspondence between the set of rationalhomotopy types of nilpotent spaces and the set of classes of isomorphic minimal dg-algebras over Q. Under this correspondence, there is a natural isomorphism

Hom(πi(X),Q

) ∼=M iX

/(M+

X ·M+X),

i.e. the rank of the ith rational homotopy group of X equals the number of generatorsof degree i in the minimal model of X.

For the proof, see [194, Theorem IV.8].

Definition B.2.6. A space X is formal if C∗(X ;Q) is a formal dg-algebra(equivalently, if APL(X) is a formal commutative dg-algebra, see Definition A.4.5).

If X in nilpotent, then it is formal if and only if there is a quasi-isomorphismMX → H∗(X ;Q). If X is a smooth manifold, then it is formal if and only if thede Rham algebra Ω∗(X) is formal; in this case we can define MX as the minimalmodel of Ω∗(X) instead of APL(X).

Example B.2.7.1. Let X = S2n+1 be an odd sphere, n > 1. Then

MX = Λ[x], deg x = 2n+ 1, dx = 0.

There is a quasi-isomorphismMX → Ω∗(X) sending x to the volume form of S2n+1.2. Let X = S2n, n > 1. Then

MX = Λ[y]⊗ R[x], deg x = 2n, deg y = 4n− 1, dx = 0, dy = x2.

The map MX → Ω∗(X) sends x to the volume form of S2n and y to zero.3. Let X = CPn, n > 1. Then

MX = Λ[y]⊗ R[x], deg x = 2, deg y = 2n+ 1, dx = 0, dy = xn+1.


There is a quasi-isomorphism MX → Ω∗(X) sending x to the Fubini–Study 2-formω = i

2π∂∂ log |z |2 ∈ Ω2(CPn) and y to zero.4. Let X = CP∞. Then

MX = Q[v] = H∗(X ;Q), deg v = 2, dv = 0.

5. Let X = CP∞ ∨ CP∞. Then

H∗(X ;Q) = Q[v1, v2]/(v1v2), deg v1 = deg v2 = 2,

MX = Q[v1, v2, w], degw = 3, dv1 = dv2 = 0, dw = v1v2,

and the map MX → H∗(X ;Q) sends w to zero. Theorem B.2.5 gives the followingnontrivial rational homotopy groups:

π2(X)⊗Q = Q2, π3(X)⊗Q = Q.

This conforms with the homotopy fibration

S3 → CP∞ ∨ CP∞ → CP∞ × CP∞

(see Example 4.3.4).

In all examples above the space X is formal (an exercise). Here is an exampleof a nonformal manifold.

Example B.2.8. Let G be the Heisenberg group consisting of matrices1 x z0 1 y0 0 1

, x, y, z ∈ R,

and let Γ = GZ be the subgroup consisting of matrices with x, y, z ∈ Z. The quotientmanifold X = G/Γ is the classifying space for the nilpotent group Γ = π1(X), i.e.X = K(Γ, 1). The minimal model MX is generated by the left-invariant forms

ω1 = dx, ω2 = dy, ω3 = xdy − dz.This dg-algebra is isomorphic to the dg-algebra of Example A.4.7. Therefore, themanifold X is not formal.

Exercises.

B.2.9. Show that all spaces of Example B.2.7 are formal.

B.3. Group actions and equivariant cohomology

There is a vast literature available on this classical subject; we mentionthe monographs of Bredon [41], Hsiang [164], Allday–Puppe [4] and Guillemin–Ginzburg–Karshon [142], among others. We briefly review some basic conceptsand results used in the main part of the book.

Let X be a Hausdorff space and G a Hausdorff topological group. One saysthat G acts on X if for any element g ∈ G there is a homeomorphism ϕg : X → X ,and the assignment g 7→ ϕg respects the algebraic and topological structure. Inmore precise terms, a (left) action of G on X is given by a continuous map

G×X → X, (g, x) 7→ gx

such that g(hx) = (gh)x for any g, h ∈ G, x ∈ X , and ex = x, where e is the unitof G. The space X is called a (left) G-space. Right actions and right G-spaces are

B.3. GROUP ACTIONS AND EQUIVARIANT COHOMOLOGY 421

defined similarly. In the case when G is an abelian group, the notions of left andright action coincide.

A continuous map f : X → Y of G-spaces is equivariant if it commutes withthe group actions, i.e. f(gx) = g · f(x) for all g ∈ G and x ∈ X . A map f is weaklyequivariant if there is an automorphism ψ : G→ G such that f(gx) = ψ(g)(f(x)) forall g ∈ G and x ∈ X . A weakly equivariant map corresponding to an automorphismψ is also referred to as ψ-equivariant.

Let x ∈ X . The set

Gx = g ∈ G : gx = xof elements of G fixing the point x is a closed subgroup in G, called the stationarysubgroup, or the stabiliser of x. The subspace

Gx = gx ∈ X : g ∈ G ⊂ Xis called the orbit of x with respect to the action of G (or the G-orbit for short). Ifpoints x and y are in the same orbit, then their stabilisers Gx and Gy are conjugatesubgroups in G. The type of an orbit Gx is the conjugation class of stabilisersubgroups of points in Gx.

The set of all orbits is denoted by X/G, and we have the canonical projectionπ : X → X/G. The space X/G with the standard quotient topology (a subsetU ⊂ X/G is open if and only if π−1(U) is open) is referred to as the orbit space, orthe quotient space. If G is a compact group, then the quotient X/G is Hausdorff,and the projection π : X → X/G is a closed and proper map (i.e. the image of aclosed subset is closed, and the preimage of a compact subset is compact).

A point x ∈ X is fixed if Gx = x, i.e. Gx = G. The set of all fixed points of aG-space X will be denoted by XG. A G-action on X is

– effective if the trivially acting subgroup g ∈ G : gx = x for all x ∈ X istrivial (consists of the single element e ∈ G);

– free if all stabilisers Gx are trivial;– almost free if all stabilisers Gx are finite subgroups of G;– semifree if any stabiliser Gx is either trivial or is the whole G;– transitive if for any two points x, y ∈ X there is an element g ∈ G such

that gx = y (i.e. X is a single orbit of the G-action).

A principal G-bundle is a locally trivial bundle p : X → B such that G acts onX preserving fibres, and the induced G-action on each fibre is free and transitive. Itfollows that each fibre is homeomorphic to G, the G-action onX is free, the G-orbitsare precisely the fibres, and the projection p : X → B induces a homeomorphismbetween the quotient X/G and the base B. Therefore p can be regarded as theprojection onto the orbit space of a free G-action. If the group G is compact, theconverse is also true under some mild topological assumptions on X : a free G-spaceX is the total space of a principal G-bundle (see [41, Chapter II]). Therefore, inthis case the notions of a principal G-bundle and a free G-action are equivalent.

Now let G be a compact Lie group. Then there exists a principal G-bundleEG → BG whose total space EG is contractible. This bundle has the followinguniversality property. Let E → B be another principal G-bundle over a cell com-plex B. Then there is a unique up to homotopy map f : B → BG such that thepullback of the bundle EG→ BG along f is the bundle E → B. The space EG isreferred to as the universal G-space, and the space BG is the classifying space forfree G-actions (or simply the classifying space for G).


Let X be a G-space. The diagonal G-action on EG×X , given by

g(e, x) = (ge, gx), g ∈ G, e ∈ EG, x ∈ X,

is free. Its orbit space is denoted by EG×GX (we shall also use the notation BGX)and is called the Borel construction, or the homotopy quotient of X by G. (Thelatter term is used since the free G-space EG ×X is homotopy equivalent to theG-space X .) There are two canonical projections

(B.4)EG×X → EG

(e, x) 7→ eand

EG×X → X

(e, x) 7→ x.

After taking quotient by the G-actions, the second projection above induces amap EG ×G X → X/G between the homotopy and ordinary quotients, which is ahomotopy equivalence when the G-action is free. The first projection gives rise toa bundle EG ×G X → BG with fibre X and structure group G, called the bundleassociated with the G-space X .

More generally, if E → B is a principal G-bundle (i.e. E is a free G-space) andX is a G-space, then we have a bundle E ×G X → B over B with fibre X . WhenX is an n-dimensional G-representation space, we obtain a vector bundle over Bwith structure group G. Real, oriented and complex n-dimensional (n-plane) vectorbundles correspond to the cases G = GL(n,R), SL(n,R) and GL(n,C), respectively.By introducing appropriate metrics, their structure groups can be reduced to O(n),SO(n) and U (n), respectively.

Now let a compact Lie group G act on a smooth manifold M by diffeomor-phisms. Given a point x ∈ M with stabiliser Gx and orbit Gx, the differential ofthe action of an element g ∈ Gx is a linear transformation of the tangent spaceTxM which is the identity on the tangent space to the orbit of x. Therefore, weobtain the induced representation of Gx in the space TxM/Tx(Gx) of the orbit,which is called the isotropy representation. In particular, when x is a fixed point(i.e. Gx = G and Gx = x), we obtain a representation of G in TxM , which is alsocalled the tangential representation of G at a fixed point x.

Theorem B.3.1 (Slice Theorem). Let a compact Lie group G act on a smoothmanifold M . Then, for each point x ∈ M , the orbit Gx has a G-invariant neigh-bourhood G-equivariantly diffeomorphic to G ×Gx

(TxM/Tx(Gx)

). The latter is a

vector bundle with fibre TxM/Tx(Gx) over G/Gx ∼= Gx and the diffeomorphismtakes the orbit Gx to the zero section of this bundle.

The slice theorem was proved by Koszul in the beginning of 1950s. Its moregeneral version for proper actions of noncompact Lie groups was proved by Palais;this proof can be found e.g. in [142, Theorem B.23] The slice theorem has manyimportant consequences. One is that the union of orbits of a given type is a (smooth,but possibly disconnected) submanifold of M . In particular, the fixed point set MG

is a submanifold. Another consequence is that the quotient M/G by a free actionof G is a smooth manifold.

The second fundamental result is the equivariant embedding theorem:

Theorem B.3.2. Let a compact Lie group G act on a compact smooth mani-fold M . Then there exist an equivariant embedding of M into a finite-dimensionallinear representation space of G.

B.3. GROUP ACTIONS AND EQUIVARIANT COHOMOLOGY 423

This theorem was proved by Mostow and Palais in 1957 (instead of compactnessof M they only assumed it to have a finite number of G-orbit types). A simple proofin the case of compact M (due to Mostow) can be found in [142, Theorem B.50].

The equivariant cohomology of X with coefficients in a ring k is defined by

H∗G(X ;k) = H∗(EG×G X ;k).

Hence, H∗G(pt ;k) = H∗(BG;k), and the projection EG ×G X → BG turns

H∗G(X ;k) into a H∗

G(pt;k)-module.For a pair of G-spaces (X,A) (where the inclusion A ⊂ X is an equivariant

map), there is a long exact sequence

· · · → HqG(X,A)

j∗−→ HqG(X)

i∗−→ HqG(A)

d−→ Hq+1G (X,A)→ · · ·

Its existence follows from the exact sequence in ordinary cohomology and the equi-variant homeomorphism (EG×X)/(EG×A) ∼= (EG× (X/A))/(EG× pt).

A bundle π : E → X with fibre F is called a G-equivariant bundle if π is anequivariant map of G-spaces. By applying the Borel construction to a G-equivariantbundle π : E → X we obtain a bundle BGE → BGX with the same fibre F . Theequivariant characteristic classes of a G-equivariant vector bundle π : E → X aredefined as the ordinary characteristic classes of the corresponding bundle BGE →BGX . For example, the equivariant Stiefel–Whitney classes of a G-equivariantvector bundle π : E → X belong to H∗

G(X ;Z2) and are denoted by wGi (E). IfE → X is an oriented G-equivariant vector bundle, then the equivariant Eulerclass eG(E) ∈ H∗

G(X ;Z) is defined. If E → X is a complex bundle and the G-action preserves the fibrewise complex structure, then the equivariant Chern classescGi (E) ∈ H2i

G (X ;Z) are defined.Now let M be a smooth oriented G-manifold of dimension n, where G is a

compact Lie group. Let N ⊂M be a G-invariant (e.g., fixed) oriented submanifoldof codimension k. We can identify the G-equivariant normal bundle ν(N ⊂M) witha G-invariant tubular neighbourhood U of N in M by means of a G-equivariantdiffeomorphism. The same diffeomorphism identifies the Thom space Th ν (seeSection D.2) of the bundle ν with the quotient space U/∂U . We have the embeddingi : N ⊂M , the projection π : U → N , and the Pontryagin–Thom map p : M → Th νcontracting the complement M \U to a point. In equivariant cohomology, the Thomclass τN ∈ Hk

G(Th ν) is uniquely determined by the identity(α · p∗(τN ), 〈M〉

)=(i∗α, 〈N〉

),

for any α ∈ Hn−kG (M). Here 〈M〉 ∈ HG

n (M) denotes the fundamental class of Min equivariant homology. The Gysin homomorphism in equivariant cohomology isdefined by the composition

H∗−kG (N)

π∗

−−−−→ H∗−kG (U)

·τN−−−−→ H∗G(Th ν)

p∗−−−−→ H∗G(M),

and is denoted by i∗. Then i∗(i∗(1)) ∈ HkG(N) is the equivariant Euler class of the

normal bundle ν(N ⊂M).

Exercises.

B.3.3. Let G be a compact group acting on a Hausdorff spaceX . Show that theG-orbits are closed, the orbit space X/G is Hausdorff, and the projection π : X →X/G is a closed and proper map.


B.3.4. If the quotient X/G is Hausdorff, then all G-orbits are closed.

B.3.5. Give an example of a Hausdorff space X with a G-action whose orbitsare closed, but the quotient X/G is not Hausdorff.

B.3.6. Show than any complex line bundle over the complex projective spaceCPn has the form S2n+1 ×S1 C. Here the S1-action on S2n+1 is standard (thediagonal action on a unit sphere in Cn+1, with quotient CPn), and C is a certain1-dimensional S1-representation space. The same line bundle can be also given as(Cn+1 \ 0)×C× C, where C× = GL (1,C).

In particular, when C is the standard (weight 1) S1-representation space (givenby S1 × C→ C, (g, z) 7→ gz), the bundle S2n+1 ×S1 C is the canonical line bundleover CPn (it is also known as the line bundle of hyperplane section and denotedby O(1) in algebraic geometry literature). The Hopf (or tautological) line bundle,whose fibre over a line ℓ ∈ CPn is ℓ itself, corresponds to the S1-representation ofweight −1, given by (g, z) 7→ g−1z.

B.3.7. Deduce from the slice theorem that if a compact Lie group G actssmoothly on a smooth manifold M , then the union of orbits of a given type (inparticular, the fixed point set MG) is a submanifold of M . Also, deduce that if theaction is free then the quotient M/G is a smooth manifold.

B.3.8. Show that if a compact Lie group G acts on M effectively and M isconnected, then the isotropy representation Gx → GL

(TxM/Tx(Gx)

)is faithful.

B.4. Eilenberg–Moore spectral sequences

In their paper [105] of 1966, Eilenberg and Moore constructed a spectral se-quence, which became one of the important computational tools of algebraic topol-ogy. It particular, it provides a method for calculation of the cohomology of thefibre of a bundle E → B using the canonical H∗(B)-module structure on H∗(E).This spectral sequence can be considered as an extension of Adams’ approach tocalculating cohomology of loop spaces [1]. In the 1960–70s applications of theEilenberg–Moore spectral sequence led to many important results on cohomology ofhomogeneous spaces for Lie groups. More recently it has been used for different cal-culations with toric spaces. This section contains the necessary information aboutthe spectral sequence; we mainly follow L. Smith’s paper [287] in this description.For a detailed account of differential homological algebra and the Eilenberg–Moorespectral sequence, as well as its applications which go beyond the scope of thisbook, we refer to McCleary’s book [215].

Here we assume that k is a field. The following theorem provides an algebraicsetup for the Eilenberg–Moore spectral sequence.

Theorem B.4.1 (Eilenberg–Moore [287, Theorem 1.2]). Let A be a differentialgraded k-algebra, and let M , N be differential graded A-modules. Then there existsa spectral sequence Er, dr converging to TorA(M,N) and whose E2-term is

E−i,j2 = Tor−i,jH[A]

(H [M ], H [N ]

), i, j > 0,

where H [ · ] denotes the algebra or module of cohomology.

Remark. The construction of Tor for differential graded objects requires someadditional considerations (see e.g. [287] or [203, Chapter XII]).

B.4. EILENBERG–MOORE SPECTRAL SEQUENCES 425

The spectral sequence of Theorem B.4.1 lives in the second quadrant and itsdifferentials dr add (r, 1 − r) to the bidegree, for r > 1. We shall refer to it as thealgebraic Eilenberg–Moore spectral sequence. Its E∞-term is expressed via a certaindecreasing filtration F−pTorA(M,N) in TorA(M,N) by the formula

E−p,n+p∞ = F−p

( ∑

−i+j=n

Tor−i,jA (M,N))/

F−p+1( ∑

−i+j=n

Tor−i,jA (M,N)).

Topological applications of Theorem B.4.1 arise in the case when A,M,N arecochain algebras of topological spaces. The classical situation is described by thecommutative diagram

(B.5)

E −−−−→ E0yy

B −−−−→ B0,

where E0 → B0 is a Serre fibre bundle with fibre F over a simply connected base B0,and E → B is the pullback along a continuous map B → B0. For any space X , letC∗(X) denote the singular k-cochain algebra of X . Then C∗(E0) and C∗(B) areC∗(B0)-modules. Under these assumptions the following statement holds.

Lemma B.4.2 ([287, Proposition 3.4]). TorC∗(B0)(C∗(E0), C

∗(B)) is a k-algebra in a natural way, and there is a canonical isomorphism of algebras

TorC∗(B0)

(C∗(E0), C

∗(B))→ H∗(E).

Applying Theorem B.4.1 in the case A = C∗(B0), M = C∗(E0), N = C∗(B)and taking into account Lemma B.4.2, we come to the following statement.

Theorem B.4.3 (Eilenberg–Moore). There exists a spectral sequence Er, drof commutative algebras converging to H∗(E) with

E−i,j2 = Tor−i,jH∗(B0)

(H∗(E0), H

∗(B)).

The spectral sequence of Theorem B.4.3 is known as the (topological) Eilenberg–Moore spectral sequence. The case when B is a point is of particular importance,and we state the corresponding result separately.

Corollary B.4.4. Let E → B be a fibration over a simply connected space Bwith fibre F . Then there exists a spectral sequence Er, dr of commutative algebrasconverging to H∗(F ) with

E2 = TorH∗(B0)

(H∗(E0),k

).

We refer to the spectral sequence of Corollary B.4.4 as the Eilenberg–Moorespectral sequence of the fibration E → B. In the case when E0 is a contractible weobtain a spectral sequence converging to the cohomology of the loop space ΩB0.

Remark. As we outlined in Section B.2, the Sullivan algebra APL(X) providesa commutative rational model for X . It can be proved [39, §3] that the aboveresults on the Eilenberg–Moore spectral sequence hold over Q with C∗ replacedby APL. This result is not a direct corollary of algebraic properties of Tor, sincethe integration map APL(X)→ C∗(X,Q) is not multiplicative.

APPENDIX C

Categorical constructions

In this appendix we introduce aspects of category theory that are directlyrelevant to the study of toric spaces. Our exposition follows closely the introductorysections of the work of Panov and Ray [260].

C.1. Diagrams and model categories

We use small capitals to denote categories. The set of morphisms betweenobjects c and d in a category c will be denoted by Morc(c, d), or simply by c(c, d).The opposite category cop has the same objects with morphisms reverted.

We shall work with the following categories of combinatorial origin:

· set: sets and set maps;· ∆: finite ordered sets [n] and nondecreasing maps;· cat(K): simplices of a finite simplicial complex K and their inclusions

(the face category of K).

Here cat(K) is an example of a more general poset category p, whose objectsare elements σ of a poset (P ,6) and there is a morphism σ → τ whenever σ 6 τ .

A category is small if both its objects and morphisms are sets. Among the threebasic categories above, ∆ and cat(K) are small, while set is not. Furthermore,cat(K) is finite.

Given a small category s and an arbitrary category c, a covariant functorD : s → c is known as an s-diagram in c. The source s is referred to as theindexing category of the diagram D. Such diagrams are themselves the objects of adiagram category [s,c], whose morphisms are natural transformations. When s is∆op, the diagrams are known as simplicial objects in c, and are written as D•; theobject D[n] is abbreviated to Dn for every n > 0, and forms the n-simplices of D•.A simplicial set is therefore a simplicial object in set, i.e. an object in the diagramcategory [∆op, set].

We may interpret every object c of c as a constant s-diagram, and so define theconstant functor κ : c→ [s,c]. Whenever κ admits a right or left adjoint [s,c]→ c,it is known as the limit or colimit functor respectively. In more detail, the limit of adiagram D : s→ c is an object limD in c for which there are natural identificationsof the morphism sets,

Mor[s,c](κ(c),D) ∼= Morc(c, limD)for any object c of c. Similarly, colimD satisfies

Morc(colimD, c) ∼= Mor[s,c](D, κ(c)).Simple examples are products and more general pullbacks, which are limits over theindexing category • → • ← •. Similarly, coproducts and more general pushouts arecolimits over the indexing category • ← • → •.

427

428 C. CATEGORICAL CONSTRUCTIONS

For any object c of c, the objects of the overcategory c ↓ c are morphismsf : b → c with fixed target, and the morphisms are the corresponding commuta-tive triangles; the full subcategory c ⇓ c is given by restricting attention to non-identities f . Similarly, the objects of the undercategory c↓c are morphisms f : c→ dwith fixed source, and the morphisms are the corresponding triangles; c⇓c is givenby restriction to the non-identities. In cat(K) for example, we have

cat(K)↓I = cat(∆(I)), cat(K)⇓I = cat(∂∆(I)),

I ↓cat(K) = cat(stK I), I ⇓cat(K) = cat(lkK I),

for any I ∈ K, where ∆(I) and ∂∆(I) denote the simplex with vertices I and itsboundary, and star and link are given in Definition 2.2.13.

A model category mc is a category which is closed with respect to formationof small limits and colimits, and contains three distinguished classes of morphisms:weak equivalencies w, fibrations p, and cofibrations i. Unless otherwise stated, theseletters denote such morphisms henceforth. A fibration or cofibration is acyclicwhenever it is also a weak equivalence. The three distinguished morphisms arerequired to satisfy the following axioms (see Hirschhorn [158, Definition 7.1.3]):

(a) a retract of a distinguished morphism is a distinguished morphism of thesame class;

(b) if f ′ · f is a composition of morphisms f and f ′, and two of the threemorphisms f , f ′ and f ′ · f is a weak equivalence, then so is the third;

(c) acyclic cofibrations obey the left lifting property with respect to fibrations,and cofibrations obey the left lifting property with respect to acyclic fi-brations;

(d) every morphism f factorises functorially as

(C.1) h = p · i = p′ · i′,for some acyclic p and i′.

These strengthen Quillen’s original axioms for a closed model category [270] intwo minor but significant ways. Quillen demanded only closure with respect tofinite limits and colimits, and only existence of factorisation (C.1) rather thanits functoriality. When using results of pioneering authors such as Bousfield andGugenheim [39] and Quillen [271], we must take account of these differences.

The axioms for a model category are actually self-dual, in the sense that anygeneral statement concerning fibrations, cofibrations, limits, and colimits is equiva-lent to the statement in which they are replaced by cofibrations, fibrations, colimits,and limits respectively. In particular, mcop always admits a dual model structure.

The axioms imply that initial and terminal objects and ∗ exist in mc, andthat mc↓M and M ↓mc inherit model structures for any object M .

An object of mc is cofibrant when the natural morphism → M is a cofibra-tion, and is fibrant when the natural morphism M → ∗ is a fibration. A cofibrantapproximation to an object N is a weak equivalence N ′ → N with cofibrant source,and a fibrant approximation is a weak equivalence N → N ′′ with fibrant target.The full subcategories mcc, mcf and mccf are defined by restricting attention tothose objects of mc that are respectively cofibrant, fibrant, and both. When appliedto → N and N → ∗, the factorisations (C.1) determine a cofibrant replacementfunctor ω : mc → mcc, and a fibrant replacement functor ϕ : mc → mcf . It fol-lows from the definitions that ω and ϕ preserve weak equivalences, and that the

C.1. DIAGRAMS AND MODEL CATEGORIES 429

associated acyclic fibrations ω(N) → N and acyclic cofibrations N → ϕ(N) formcofibrant and fibrant approximations respectively. These ideas are central to thedefinition of homotopy limits and colimits given in Section C.3 below.

Weak equivalences need not be invertible, so objects M and N are deemed

to be weakly equivalent if they are linked by a zigzag Me1←− · · · en−→ N in mc;

this is the smallest equivalence relation generated by the weak equivalences. Animportant consequence of the axioms is the existence of a localisation functorγ : mc → Ho(mc), such that γ(w) is an isomorphism in the homotopy categoryHo(mc) for every weak equivalence w (i.e. Ho(mc) is obtained from mc by in-verting all weak equivalences). Here Ho(mc) has the same objects as mc, and isequivalent to a category whose objects are those of mccf , but whose morphisms arehomotopy classes of morphisms between them.

Any functor F of model categories that preserves weak equivalences induces afunctor Ho(F ) on their homotopy categories. Examples include

(C.2) Ho(ω) : Ho(mc)→ Ho(mcc) and Ho(ϕ) : Ho(mc)→ Ho(mcf ).

Such functors often occur as adjoint pairs

(C.3) F : mb mc :G ,

where F is left Quillen if it preserves cofibrations and acyclic cofibrations, and G isright Quillen if it preserves fibrations and acyclic fibrations. Either of these impliesthe other, leading to the notion of a Quillen pair (F,G); then Ken Brown’s Lemma[158, Lemma 7.7.1] applies to show that F and G preserve all weak equivalenceson mbc and mcf respectively. So they may be combined with (C.2) to produce anadjoint pair of derived functors

LF : Ho(mb) Ho(mc) :RG,

which are equivalences of the homotopy categories (or certain of their full subcate-gories) in favourable cases.

Our first examples of model categories are of topological origin, as follows:

· top: pointed topological spaces and pointed continuous maps;· tmon: topological monoids and continuous homomorphisms;· ssets: simplicial sets.

Homotopy theorists often impose restrictions on topological spaces defining thecategory top, ensuring that it behaves nicely with respect to formation of limits andcolimits, etc. For example, top is often defined to consist of compactly generatedHausdorff spaces (a space X is compactly generated if a subset A is closed wheneverthe intersection of A with any compact subset ofX is closed), or k-spaces [315]. Weignore this subtlety however, as spaces we work with will be nice enough anyway.

There is a model structure on top in which fibrations are Hurewicz fibrations,cofibrations are defined by the homotopy extension property (B.3), and weak equiv-alences are homotopy equivalences. However, in the more convenient and standardmodel structure on top, weak equivalences are maps inducing isomorphisms of ho-motopy groups, fibrations are Serre fibrations, and cofibrations obey the left liftingproperty with respect to acyclic fibrations (this is a narrower class than maps obey-ing the the HEP (B.3)). The axioms for this model structure on top are verifiedin [163, Theorem 2.4.23], for example.

In either of the model structures above, cell complexes are cofibrant objectsin top. Recall that a cell complex can be defined as a result of iterating the


operation of attaching a cell, i.e. pushing out the standard cofibration Sn−1 → Dn,see (B.1). In the second (standard) model structure on top, every space X has acellular model, i.e. there is a weak equivalence W → X with W a cell complex,providing a cofibrant approximation. Two weakly equivalent topological spaces havehomotopy equivalent cellular models. Cellular models are not functorial, however.A genuine cofibrant replacement functor ω(X)→ X must be constructed with care,and is defined in [102, §98], for example.

A topological monoid is a space with a continuous associative product and iden-tity element. We assume that topological monoids are pointed by their identities,so that tmon is a subcategory of top. The model structure for tmon is originallydue to Schwanzl and Vogt [281], and may also be deduced from Schwede and Ship-ley’s theory [283] of monoids in monoidal model categories; weak equivalences andfibrations are those homomorphisms which are weak equivalences and fibrations intop, and cofibrations obey the appropriate lifting property.

In the standard model structure on sset, weak equivalences are maps of sim-plicial sets whose realisations are weak equivalences of spaces, fibrations are Kanfibrations (whose realisations are Serre fibrations), and cofibrations are monomor-phisms of simplicial sets. There is a Quillen equivalence

| · | : sset top :S• ,

where | · | denotes the geometric realisation of a simplicial set, and S• is the totalsingular complex of a space. It induces an equivalence of homotopy categories ofsimplicial sets and topological spaces.

Our algebraic categories are defined over arbitrary commutative rings k, buttend only to acquire model structures when k is a field of characteristic zero.

· chk and cochk: augmented chain and cochain complexes;· cdgak: commutative augmented differential graded algebras, with coho-

mology differential (raising the degree by 1);· cdgck: cocommutative coaugmented differential graded coalgebras, with

homology differential (lowering the degree by 1);· dgak: augmented differential graded algebras, with homology differential;· dgck: coaugmented differential graded coalgebras, with homology differ-

ential;· dgl: differential graded Lie algebras over Q, with homology differential.

For any model structure on these categories, weak equivalences are the quasi-isomorphisms, which induce isomorphisms in homology or cohomology. The fi-brations and cofibrations are described in Section C.2 below. The augmentationsand coaugmentations act as algebraic analogues of basepoints.

We reserve the notation amc for any of the algebraic model categories above,and assume that objects are graded over the nonnegative integers. We denote thefull subcategory of connected objects by amc0. In order to emphasise the differ-ential, we may display an object M as (M,d). The (co)homology group H(M,d)is also an R-module, and inherits all structure on M except for the differential.Nevertheless, we may interpret any graded algebra, coalgebra or Lie algebra as anobject of the corresponding differential category, by imposing d = 0.

Extending Definition A.4.5, we refer to an object (M,d) is formal in amcwhenever there exists a zigzag of quasi-isomorphisms

(C.4) (M,d) =M1≃←− · · · ≃−→Mk = (H(M), 0).

C.1. DIAGRAMS AND MODEL CATEGORIES 431

Formality only has meaning in an algebraic model category.There is special class of cofibrant objects in cdgaQ, which are analogous to

cell complexes in top. These are minimal dg-algebras, see Definition A.4.1. Anyminimal dg-algebra can be constructed by successive pushouts of the form

(C.5)

SQ(x)f−−−−→ (A, dA)yj

y

SQ(w, dw) −−−−→ B = (A⊗ S(w), dB)in a way similar to constructing cell complexes by pushouts (B.1). Here SQ(x)denotes a free commutative dg-algebra with one generator x of positive degree andzero differential, so that SQ(x) is the exterior algebra ΛQ[x] when deg x is odd andthe polynomial algebra Q[x] when deg x is even. The dg-algebra SQ(x, dx) has zerocohomology. The map j is defined by j(x) = dw. The differential in the pushoutdg-algebra B ∼= A⊗ SQ(w) is given by

dB(a⊗ 1) = dAa⊗ 1, dB(1⊗ w) = f(x)⊗ 1.

Theorem A.4.3 asserts the existence of a cofibrant approximation f : MA → A for ahomologically connected dg-algebraA, whereMA is a minimal model for A; any twominimal models for A are necessarily isomorphic, and MA and MB are isomorphicfor quasi-isomorphic A and B. The advantage of MA is that it simplifies manycalculations concerning A; disadvantages include the fact that it may be difficultto describe for relatively straightforward objects A, and that it cannot be chosenfunctorially. A genuine cofibrant replacement functor requires additional care, andseems first to have been made explicit in [39, §4.7].

Sullivan’s approach to rational homotopy theory is based on the PL-cochainfunctor APL : top → cdgaQ. Basic results related to this approach are given inSection B.2. Following [112], APL(X) is defined as A∗(S•X), where S•(X) denotesthe total singular complex of X and A∗ : sset→ cdgaQ is the polynomial de Rhamfunctor of [39]. The PL-de Rham Theorem (Theorem B.2.3) yields a natural isomor-phism H(APL(X)) → H∗(X,Q), so APL(X) provides a commutative replacementfor rational singular cochains, and APL descends to homotopy categories. Bousfieldand Gugenheim prove that it restricts to an equivalence of appropriate full subcat-egories of Ho(top) and Ho(cdgaQ). In other words, it provides a contravariantalgebraic model for the rational homotopy theory of well-behaved spaces.

Quillen’s approach involves the homotopy groups π∗(ΩX) ⊗Z Q, which formthe rational homotopy Lie algebra of X under the Samelson product . He constructsa covariant functor Q : top→ dgl, and a natural isomorphism

H [Q(X)]∼=−→ π∗(ΩX)⊗Z Q.

for any simply connected X . He concludes that Q passes to an equivalence of homo-topy categories; in other words, its derived functor provides a covariant algebraicmodel for the rational homotopy theory of simply connected spaces.

We recall from Definition B.2.6 that a spaceX is formal when APL(X) is formalin cdgaQ. A space X is referred to as coformal when Q(X) is formal in dgl.

The importance of categories of simplicial objects is due in part to the structureof the indexing category ∆op. Every object [n] has degree n, and every morphismmay be factored uniquely as a composition of morphisms that raise and lower degree.


These properties are formalised in the notion of a Reedy category a, which admitsgenerating subcategories a+ and a− whose non-identity morphisms raise and lowerdegree respectively. The diagram category [a,mc] then supports a canonical modelstructure of its own [158, Theorem 15.3.4]. By duality, aop is also Reedy, with(aop)+ = (a−)

op and vice-versa. A simple example is provided by cat(K), whosedegree function assigns the dimension |I| − 1 to each simplex I of K. So cat+(K)is the same as cat(K), and cat−(K) consists entirely of identities.

In the Reedy model structure on [cat(K),mc] , weak equivalences w : C → Dare given objectwise, in the sense that w(I) : C(I)→ D(I) is a weak equivalence inmc for every I ∈ K. Fibrations are also objectwise. To describe the cofibrations,we restrict C and D to the overcategories cat(K)⇓I = cat(∂∆(I)), and write LICand LID for their respective colimits. So LI is the latching functor of [163], andg : C → D is a cofibration precisely when the induced maps

(C.6) C(I) ∐LIC LID −→ D(I)

are cofibrations in mc for all I ∈ K. Thus D : cat(K)→ mc is cofibrant when everycanonical map colimD|

cat(∂∆(I)) → D(I) is a cofibration.In the dual model structure on [catop(K),mc] , weak equivalences and cofibra-

tions are given objectwise. To describe the fibrations, we restrict C and D to theundercategories catop(∂∆(I)), and write MIC and MID for their respective limits.So MI is the matching functor of [163], and f : C → D is a fibration precisely whenthe induced maps

(C.7) C(I) −→ D(I)×MID MIC

are fibrations in mc for all I ∈ K. Thus C : catop(K) → mc is fibrant when everycanonical map C(I)→ lim C|

catop(∂∆(I)) is a fibration.

C.2. Algebraic model categories

Here we give further details of the algebraic model categories introduced inthe previous section. We describe the fibrations and cofibrations in each category,comment on the status of the strengthened axioms, and give simple examples inless familiar cases. We also discuss two important adjoint pairs.

So far as general algebraic notation is concerned, we work over an arbitrarycommutative ring k. We indicate the coefficient ring k by means of a subscriptwhen necessary, but often omit it. In some situations k is restricted to the rationalnumbers Q, in this case we always clearly indicate it in the notation.

We consider finite setsW of generators w1, . . . , wm. We write the graded tensork-algebra on W as Tk(w1, . . . , wm), and use the abbreviation Tk(W) wheneverpossible. Its symmetrisation Sk(W) is the graded commutative k-algebra generatedby W . If U , V ⊂ W are the subsets of odd and even grading respectively, thenSk(W) is the tensor product of the exterior algebra Λk[U ] and the polynomialalgebra k[V ]. When k is a field of characteristic zero, it is also convenient to denotethe free graded Lie algebra onW and its commutative counterpart by FLk(W) andCLk(W) respectively; the latter is nothing more than a free k-module.

Almost all of our graded algebras have finite type, leading to a natural coalge-braic structure on their duals. We write the free tensor coalgebra on W as Tk〈W〉;

C.2. ALGEBRAIC MODEL CATEGORIES 433

it is isomorphic to Tk(W) as k-modules, and its diagonal is given by

∆(wj1 ⊗ · · · ⊗ wjr ) =r∑

k=0

(wj1 ⊗ · · · ⊗ wjk)⊗ (wjk+1⊗ · · · ⊗ wjr ).

The submodule Sk〈W〉 of symmetric elements (wi ⊗ wj + (−1)degwi degwjwj ⊗ wi,for example) is the graded cocommutative k-coalgebra cogenerated by W .

Given W , we may sometimes define a differential by denoting the set of ele-ments dw1, . . . , dwm by dW . For example, we write the free dg-algebra on a singlegenerator w of positive dimension as Tk(w, dw); the notation is designed to reinforcethe fact that its underlying algebra is the tensor k-algebra on elements w and dw.Similarly, Tk〈w, dw〉 is the free differential graded coalgebra on w. For furtherinformation on differential graded coalgebras, [166] remains a valuable source.

Chain and cochain complexes. The existence of a model structure on cat-egories of chain complexes was first proposed by Quillen [270], whose view of ho-mological algebra as homotopy theory in chk was a crucial insight. Variationsinvolving bounded and unbounded complexes are studied by Hovey [163], for ex-ample. In chk, we assume that the fibrations are epimorphic in positive degreesand the cofibrations are monomorphic with degree-wise projective cokernel [102].In particular, every object is fibrant.

The existence of limits and colimits is assured by working dimensionwise, andfunctoriality of the factorisations (C.1) follows automatically from the fact that chk

is cofibrantly generated [163, Chapter 2].Model structures on cochk are established by analogous techniques. It is

usual to assume that the fibrations are epimorphic with degree-wise injective kernel,and the cofibrations are monomorphic in positive degrees. Then every object iscofibrant. There is an alternative structure based on projectives, but we shall onlyrefer to the rational case so we ignore the distinction.

Tensor product of (co)chain complexes invests chk and cochk with the struc-ture of a monoidal model category, as defined by Schwede and Shipley [283].

Commutative differential graded algebras. We consider commutative dif-ferential graded algebras over Q with cohomology differentials, so they are commu-tative monoids in cochQ. A model structure on cdgaQ was first defined in thiscontext by Bousfield and Gugenheim [39], and has played a significant role in thetheoretical development of rational homotopy theory ever since. The fibrations areepimorphic, and the cofibrations are determined by the appropriate lifting property;some care is required to identify sufficiently many explicit cofibrations.

Limits in cdgaQ are created in the underlying category cochQ and endowedwith the natural algebra structure, whereas colimits exist because cdgaQ has finitecoproducts and filtered colimits. The proof of the factorisation axioms in [39] isalready functorial.

By way of example, we note that the product of algebras A and B is theiraugmented sum A⊕B, defined by pulling back the diagram of augmentations,

A⊕B −−−−→ Ay

yεA

BεB−−−−→ Q


in coch and imposing the standard multiplication on the result. The coproductis their tensor product A⊗B over Q. Examples of cofibrations include extensionsA → (A ⊗ S(w), d) given by (C.5); such an extension is determined by a cocyclez = f(x) in A. This illustrates the fact that the pushout of a cofibration is acofibration. A larger class of cofibrations A→ A⊗ S(W) is given by iteration, forany set W of positive dimensional generators corresponding to cocycles in A.

The factorisations (C.1) are only valid over fields of characteristic 0, so themodel structure does not extend to cdgak for arbitrary rings k.

Differential graded algebras. Our differential graded algebras have homol-ogy differentials, and are the monoids in chk. A model category structure in dgak istherefore induced by applying Quillen’s path object argument, as in [283]; a similarstructure was first proposed by Jardine [176] (albeit with cohomology differentials),who proceeds by modifying the methods of [39]. Fibrations are epimorphisms, andcofibrations are determined by the appropriate lifting property.

Limits are created in chk, whereas colimits exist because dgak has finite co-products and filtered colimits. Functoriality of the factorisations follows by adapt-ing the proofs of [39], and works over arbitrary k.

For example, the coproduct of algebras A and B is the free product A ⋆ B,formed by factoring out an appropriate differential graded ideal [176] from the free(tensor) algebra Tk(A⊗B) on the chain complex A⊗B. Examples of cofibrationsinclude the extensions A→ (A⋆Tk(w), d), determined by cycles z in A. By analogywith the commutative case, such an extension is defined by the pushout diagram

Tk(x)f−−−−→ A

yjy

Tk(w, dw) −−−−→ A ⋆ Tk(w)

where f(x) = z and j(x) = dw. The differential on A ⋆ Tk(w) is given by

d(a ⋆ 1) = dAa ⋆ 1, d(1 ⋆ w) = f(x) ⋆ 1.

Further cofibrations A → A ⋆ Tk(W) arise by iteration, for any set W of positivedimensional generators corresponding to cycles in A.

Cocommutative differential graded coalgebras. The cocommutativecomonoids in chk are the objects of cdgck, and the morphisms preserve co-multiplication. The model structure is defined only over fields of characteristic 0;in view of our applications, we shall restrict attention to the case Q. In practice,we interpret cdgcQ as the full subcategory cdgc0,Q of connected objects C, whichare necessarily coaugmented. Model structure was first defined on the category ofsimply connected rational cocommutative coalgebras by Quillen [271], and refinedto cdgc0,Q by Neisendorfer [239]. The cofibrations are monomorphisms, and thefibrations are determined by the appropriate lifting property.

Limits exist because cdgcQ has finite products and filtered limits, whereascolimits are created in chQ, and endowed with the natural coalgebra structure.Functoriality of the factorisations again follows by adapting the proofs of [39].

For example, the product of coalgebras C and D is their tensor product C⊗Dover Q. The coproduct is their coaugmented sum, given by pushing out the diagram


of coaugmentations

QδC−−−−→ C

δD

yy

D −−−−→ C ⊕Din chQ and imposing the standard comultiplication on the result. Examples offibrations include the projections (C ⊗ SQ〈dt〉, d) → C, which are determined bycycles z in C and defined by the pullback diagram

C ⊗ S〈dt〉 −−−−→ S〈t, dt〉y

yq

Ch−−−−→ S〈x〉

where q(t) = x, q(dt) = 0 and h(z) = x. The differential on C ⊗ SQ〈dt〉 satisfies

d(z ⊗ 1) = 1⊗ dt, d(1 ⊗ dt) = 0.

This illustrates the fact that the pullback of a fibration is a fibration. Furtherfibrations C ⊗ SQ〈dT 〉 → C are given by iteration, for any set T of generatorscorresponding to elements of degree > 2 in C.

Differential graded coalgebras. Model structures on more general cate-gories of differential graded coalgebras have been publicised by Getzler and Go-erss [126], who also work over a field. Once again, we restrict attention to Q. Theobjects of dgcQ are comonoids in chQ, and the morphisms preserve comultiplica-tion. The cofibrations are monomorphisms, and the fibrations are determined bythe appropriate lifting property.

Limits exist because dgcQ has finite products and filtered limits, and colimitsare created in chQ. Functoriality of factorisations follows from the fact that themodel structure is cofibrantly generated.

For example, the product of coalgebras C and D is the cofree product C ⋆ D[126]. Their coproduct is the coaugmented sum, as in the case of cdgcQ. Examplesof fibrations include the projections [C ⋆ TQ〈dt〉, d] → C, which are determined bycycles z in C and defined by the pullback diagram

C ⋆ TQ〈dt〉 −−−−→ TQ〈t, dt〉yyq

Ch−−−−→ TQ〈x〉

where q(t) = x, q(dt) = 0 and h(z) = x.

Differential graded Lie algebras. A (rational) differential graded Lie alge-bra L is a chain complex in chQ, equipped with a bracket morphism [ , ] : L⊗L→ Lsatisfying signed versions of the antisymmetry and Jacobi identity:

(C.8) [x, y] = −(−1)degx deg y[y, x], [x, [y, z]] = [[x, y], z]+(−1)degx deg y[y, [x, z]].

Differential graded Lie algebras over Q are the objects of the cagegory dgl. Quillen[271] originally defined a model structure on the subcategory of reduced objects,which was extended to dgl by Neisendorfer [239]. Fibrations are epimorphisms,and cofibrations are determined by the appropriate lifting property.


Limits are created in chQ, whereas colimits exist because dgl has finite coprod-ucts and filtered colimits. Functoriality of the factorisations follows by adaptingthe proofs of [239].

For example, the product of Lie algebras L and M is their product L ⊕M aschain complexes, with the induced bracket structure. Their coproduct is the freeproduct L ⋆ M , obtained by factoring out an appropriate differential graded idealfrom the free Lie algebra FL(L ⊗M) on the chain complex L ⊗M . Examples ofcofibrations include the extensions L → (L ⋆ FL(w), d), which are determined bycycles z in L and defined by the pushout diagram

FL(x)f−−−−→ L

yjy

FL(w, dw) −−−−→ L ⋆ FL(w)

where f(x) = z and j(x) = dw. The differential on L ⋆ FL(w) is given by

d(l ⋆ 1) = dLl ⋆ 1, d(1 ⋆ w) = z ⋆ 1.

For historical reasons, a differential graded Lie algebra L is said to be coformalwhenever it is formal in dgl.

Adjoint pairs. Following Moore [234], [166], we consider the algebraic clas-sifying functor B∗ and the loop functor Ω∗ as an adjoint pair

(C.9) Ω∗ : dgc0,k dgak :B∗.

For any object A of dgak, the classifying coalgebra B∗A agrees with Eilenbergand Mac Lane’s normalised bar construction as objects of chk. For any objectC of dgc0,k, the loop algebra Ω∗C is given by the tensor algebra Tk(s

−1C) on

the desuspended k-module C = Ker(ε : C → k), and agrees with Adams’ cobarconstruction [1] as objects of chk.

The classical result of Adams links the Moore loop functor Ω : top → tmonwith its algebraic analogue Ω∗:

Theorem C.2.1 ([1]). For a simply connected pointed space X and a commu-tative ring k, there is a natural isomorphism of graded algebras

H(Ω∗C∗(X ;k)) ∼= H∗(ΩX ;k),

where C∗(X ;k) denotes the suitably reduced singular chain complex of X.

The isomorphism of Theorem C.2.1 is induced by a natural homomorphism

Ω∗C∗(X ;k) −→ CU ∗(ΩX ;k)

of dgak, where CU ∗(ΩX ;k) denotes the suitably reduced cubical chains on ΩXwith the dg-algebra structure induced from composition of Moore loops.

The graded homology algebra H(Ω∗C) is denoted by CotorC(k,k). When k isa field, there is an isomorphism

(C.10) CotorC(k,k) ∼= ExtC∗(k,k)

of graded algebras [267, page 41], where C∗ is the graded algebra dual to C andExtC∗(k,k) is the Yoneda algebra of C∗ [203].


Proposition C.2.2. The loop functor Ω∗ preserves cofibrations of connectedcoalgebras and weak equivalences of simply connected coalgebras; the classifyingfunctor B∗ preserves fibrations of connected algebras and all weak equivalences.

Proof. The fact that B∗ and Ω∗ preserve weak equivalences of algebras andsimply connected coalgebras respectively is proved by standard arguments with theEilenberg–Moore spectral sequence [111, page 538]. The additional assumptionfor coalgebras is necessary to ensure that the cobar spectral sequence converges,because the relevant filtration is decreasing.

Given any cofibration i : C1 → C2 of connected coalgebras, we must checkthat Ω∗i : Ω∗C1 → Ω∗C2 satisfies the left lifting property with respect to anyacyclic fibration p : A1 → A2 in dgak. This involves finding lifts Ω∗C2 → A1 andC2 → B∗A1 in the respective diagrams

Ω∗C1 −−−−→ A1

Ω∗i

yyp

Ω∗C2 −−−−→ A2

and

C1 −−−−→ B∗A1

i

yyB∗p

C2 −−−−→ B∗A2

;

each lift implies the other, by adjointness. Since p is an acyclic fibration, its kernelA satisfiesH(A) ∼= k. In this circumstances, the projection B∗p splits by [166, The-orem IV.2.5], so B∗A1 is isomorphic to the cofree product B∗A2 ⋆ B∗A. Therefore,B∗p is an acyclic fibration in dgc0,k, and our lift is assured.

A second application of adjointness shows that B∗ preserves all fibrations ofconnected algebras.

Remark. It follows from Proposition C.2.2 that the restriction of (C.9) tosimply connected coalgebras and connected algebras respectively,

Ω∗ : dgc1,k dga0,k :B∗,

acts as a Quillen pair, and induces an adjoint pair of equivalences on appropri-ate full subcategories of the homotopy categories. An example is given in [111,p. 538] which shows that Ω∗ fails to preserve quasi-isomorphisms (or even acycliccofibrations) if the coalgebras are not simply connected.

Over Q, the adjunction maps C 7→ B∗Ω∗C and Ω∗B∗A 7→ A are quasi-isomor-phisms for any objects A and C.

Following Neisendorfer [239, Proposition 7.2], we consider a second pair ofadjoint functors

(C.11) L∗ : cdgc0,Q dgl :M∗,

whose derived functors induce an equivalence between Ho(cdgc0,Q) and a certainfull subcategory of Ho(dgl). This extends Quillen’s original results [271] for L∗ andM∗, which apply only to simply connected coalgebras and connected Lie algebras.Given a connected cocommutative coalgebra C, the underlying graded Lie algebraof L∗C is the free Lie algebra FL(s−1C) ⊂ T (s−1C). This is preserved by thedifferential in Ω∗C because C is cocommutative, thereby identifying L∗C as thedifferential graded Lie algebra of primitives in Ω∗C. The right adjoint functor M∗

may be regarded as a generalisation to differential graded objects of the standardcomplex for calculating the cohomology of Lie algebras. Given any L in dgl theunderlying cocommutative coalgebra of M∗L is the symmetric coalgebra C(sL) onthe suspended vector space L.


The ordinary (topological) classifying space functor B : top→ tmon and theMoore loop functor Ω : tmon → top are not formally adjoint, because Ω doesnot preserve products. However, as it was shown by Vogt [317], after passing toappropriate localisations, Ω becomes right adjoint to B in the homotopy categories.

There is also a similar result in simplicial category: the loop functor fromsimplicial sets to simplicial groups is left adjoint to the classifying functor, as inthe case of algebraic functors Ω∗ and B∗.

C.3. Homotopy limits and colimits

The lim and colim functors [a,mc] → mc do not generally preserve weak equiv-alences, and the theory of homotopy limits and colimits has been developed to rem-edy this deficiency. We outline their construction in this section, and discuss basicproperties.

With cat(K) and catop(K) in mind as primary examples, we assume through-out that a is a finite Reedy category.

A Reedy category a has cofibrant constants if the constant a-diagram M iscofibrant in [a,mc] , for any cofibrant object M of an arbitrary model category mc.Similarly, a has fibrant constants if the constant a-diagram N is fibrant for anyfibrant object N of mc.

As shown in [158, Theorem 15.10.8], a Reedy category a has fibrant constantsif and only if the first pair of adjoint functors

(C.12) colim: [a,mc] mc :κ, κ : mc [a,mc] : lim

is a Quillen pair (i.e. colim is left Quillen) for every model category mc. Similarly,a has cofibrant constants if and only if the second pair above is a Quillen pair, i.e.lim is right Quillen.

We now apply the fibrant and cofibrant replacement functors associated to theReedy model structure on [a,mc] , and their homotopy functors (C.2).

Definition C.3.1. For any Reedy category a with fibrant and cofibrant con-stants, and any model category mc:

(a) the homotopy colimit functor is the composition

hocolim: Ho [a,mc]Ho(ω)−−−−→ Ho [a,mc]c

Ho(colim)−−−−−→ Ho(mc) ;

(b) the homotopy limit functor is the composition

holim: Ho [a,mc]Ho(ϕ)−−−−→ Ho [a,mc]f

Ho(lim)−−−−−→ Ho(mc).

Remark. Definition C.3.1 incorporates the fact that holim and hocolim mapobjectwise weak equivalences of diagrams to weak equivalences in mc.

The Reedy categories cat(K) and catop(K) satisfy the criteria of [158, Propo-sition 15.10.2] and therefore have fibrant and cofibrant constants, for every simpli-cial complex K. This implies that holim and hocolim: Ho [cat(K),mc] → Ho(mc)are defined; and similarly for catop(K).

Describing explicit models for homotopy limits and colimits has been a majorobjective for homotopy theorists since their study was initiated by Bousfield andKan [40] and Vogt [316]. In terms of Definition C.3.1, the issue is to choose fibrantand cofibrant replacement functors ϕ and ω. Many alternatives exist, includingthose defined by the two-sided bar and cobar constructions of [261] or the frames

C.3. HOMOTOPY LIMITS AND COLIMITS 439

of [158, §16.6], but no single description yet appears to be convenient in all cases.Instead, we accept a variety of possibilities, which are often implicit; the next fewresults ensure that they are as compatible and well-behaved as we need.

Proposition C.3.2. Any cofibrant approximation D′ ≃−→ D of diagrams in-

duces a weak equivalence colimD′ ≃−→ hocolimD in mc; and any fibrant approxi-

mation D ≃−→ D′′ induces a weak equivalence holimD ≃−→ limD′′.

Proof. Using the left lifting property (axiom (c)) of the cofibration → D′

with respect to the acyclic fibration ω(D) → D we obtain a factorisation D′ →ω(D)→ D, in which the left hand map is a weak equivalence by axiom (b). But D′

and ω(D) are cofibrant, and colim is left Quillen, so the induced map colimD′ →colimω(D) is a weak equivalence, as required. The proof for lim is dual.

Remark. Such arguments may be strengthened to include uniqueness state-ments, and show that the replacements ϕ(D) and ω(D) are themselves unique upto homotopy equivalence over D, see [158, Proposition 8.1.8].

Proposition C.3.3. For any cofibrant diagram D and fibrant diagram E, there

are natural weak equivalences hocolimD ≃−→ colimD and lim E ≃−→ holimE.Proof. For D, it suffices to apply the left Quillen functor colim to the acyclic

fibration ω(D)→ D. The proof for E is dual.

Proposition C.3.4. A weak equivalence D′ ≃−→ D of cofibrant diagrams in-

duces a weak equivalence colimD′ ≃−→ colimD, and a weak equivalence E ≃−→ E ′ of

fibrant diagrams induces a weak equivalence lim E ≃−→ lim E ′.Proof. This follows from Propositions C.3.2 and C.3.3.

Proposition C.3.5. In any model category mc:

(a) if all three objects of a pushout diagram D : L ← M → N are cofibrant,and either of the maps is a cofibration, then there exists a weak equivalence

hocolimD ≃−→ colimD;(b) if all three objects of a pullback diagram E : P → Q ← R are fibrant,

and either of the maps is a fibration, then there exists a weak equivalence

lim E ≃−→ holim E.Proof. For (a), assume that M → N is a cofibration, and that the indexing

category b for D has non-identity morphisms λ ← µ → ν. The degree functiondeg(λ) = 0, deg(µ) = 1, and deg(ν) = 2 turns b into a Reedy category with fibrantconstants, and ensures that D is cofibrant. So Proposition C.3.3 applies. If M → Lis a cofibration, the corresponding argument holds by symmetry.

For (b), the proofs are dual.

There is an important situation when the homotopy colimit over a poset cate-gory can be described explicitly:

Lemma C.3.6 (Wedge Lemma [321, Lemma 4.9]). Let (P ,6) be a poset with

initial element 0, and let p be the corresponding poset category. Suppose there isdiagram D : p→ top of spaces so that D(0) = pt and D(σ)→ D(τ) is the constantmap to the basepoint for all σ < τ in P. Then there is a homotopy equivalence

hocolimD ≃−→∨

σ∈P

(| ord(P>σ)| ∗D(σ)

),


where | ord(P>σ)| is the geometric realisation of the order complex of the uppersemi-interval P>σ = τ ∈ P : τ > σ.

According to a result of Panov, Ray and Vogt, the classifying space functorB : tmon → top commutes with homotopy colimits (of topological monoids andtopological spaces, respectively) in the following sense:

Theorem C.3.7 ([261, Theorem 7.12, Proposition 7.15]). For any diagramD : a → tmon of well-pointed topological monoids with the homotopy types of cellcomplexes, there is a natural homotopy equivalence

g : hocolimtopBD ≃−→ B hocolimtmonD.Furthermore, there is a commutative square

(C.13)

hocolimtopBD ≃−−−−→ B hocolimtmonDyptop

yBptmon

colimtopBD −−−−→ B colimtmonDwhere ptop and ptmon are the natural projections.

A weaker version of the theorem above can be stated for the Moore loop functor:

Corollary C.3.8. For D : a→ tmon as above, there is a commutative square

Ω hocolimtopBD ≃−−−−→ hocolimtmonDyΩptop

yptmon

Ω colimtopBD −−−−→ colimtmonDin Ho(tmon), where the upper homomorphism is a homotopy equivalence.

Proof. This follows by applying Ω to (C.13) and then composing the hor-izontal maps with the canonical weak equivalence ΩBG → G in tmon, whereG = hocolimtmonD and colimtmonD respectively.

The lower map in the diagram above is not a homotopy equivalence in general,although Ω preserves coproducts. Appropriate examples are given in Section 8.4.

APPENDIX D

Bordism and cobordism

Here we summarise the required facts from the theory of bordism and cobor-dism, with the most attention given to complex (co)bordism. Cobordism theoryis one of the deepest and most influential parts of algebraic topology, which expe-rienced a spectacular development in the 1960s. Proofs of results presented herewould require a separate monograph and a substantial background in algebraictopology. There are exceptions where the proofs are concise and included here. Forthe rest an interested reader is referred to the works of Novikov [246], [248], andmonographs of Conner–Floyd [81], [82] and Stong [297].

We consider topological spaces which have the homotopy type of cell complexes.All manifolds are assumed to be smooth, compact and closed (without boundary),unless otherwise specified.

D.1. Bordism of manifolds

Given two n-dimensional manifolds M0 and M1, a bordism between them isan (n+1)-dimensional manifold W with boundary, whose boundary is the disjointunion of M0 and M1, that is, ∂W =M0 ⊔M1. If such a W exists, M0 and M1 arecalled bordant. The bordism relation splits the set of manifolds into equivalenceclasses (see Fig. D.1.1), which are called bordism classes.

We denote the bordism class of M by [M ], and denote by ΩOn the set of bordismclasses of n-dimensional manifolds. Then ΩOn is an Abelian group with respectto the disjoint union operation: [M1] + [M2] = [M1 ⊔ M2]. Zero is representedby the bordism class of the empty set (which is counted as a manifold in anydimension), or by the bordism class of any manifold which bounds. We also have∂(M × I) =M ⊔M . Hence, 2[M ] = 0 and ΩOn is a 2-torsion group.

Set ΩO =⊕

n>0ΩOn . The direct product of manifolds induces a multiplication

of bordism classes, namely [M1] × [M2] = [M1 × M2]. It makes ΩO a gradedcommutative ring, the unoriented bordism ring.

For any space X the bordism relation can be extended to maps of manifoldsto X : two maps M1 → X and M2 → X are bordant if there is a bordism Wbetween M1 and M2 and the map M1 ⊔M2 → X extends to a map W → X . Theset of bordism classes of maps M → X with dimM = n forms an abelian groupcalled the n-dimensional unoriented bordism group of X and denoted On(X) (othernotations: Nn(X), MOn(X)). We note that On(pt) = ΩOn , where pt is a point.

The bordism group On(X,A) of a pair A ⊂ X is defined as the set of bordismclasses of maps of manifolds with boundary, (M,∂M)→ (X,A), where dimM = n.(Two such maps f0 : (M0, ∂M0)→ (X,A) and f1 : (M1, ∂M1)→ (X,A) are bordant

Figure D.1. Transitivity of the bordism relation.

441

442 D. BORDISM AND COBORDISM

if there is W such that ∂W =M0 ∪M1 ∪M , where M is a bordism between ∂M0

and ∂M1, and a map f : W → X such that f |M0 = f0, f |M1 = f1 and f(M) ⊂ A.)We have On(X,∅) = On(X).

There is an obviously defined map ΩOm×On(X)→ Om+n(X) turning O∗(X) =⊕n>0On(X) into a graded ΩO-module. The assignment X 7→ O∗(X) defines a

generalised homology theory, that is, it is functorial in X , homotopy invariant, hasthe excision property and exact sequences of pairs.

D.2. Thom spaces and cobordism functors

A remarkable geometric construction due to Pontryagin and Thom reducesthe calculation of the bordism groups On(X) to a homotopical problem. Here weassume known basic facts from the theory of vector bundles.

Given an n-dimensional real Euclidean vector bundle ξ with total space E = Eξand Hausdorff compact base X , the Thom space of ξ is defined as the quotient

Th ξ = E/E>1,

where E>1 is the subspace consisting of vectors of length > 1 in the fibres of ξ.Equivalently, Th ξ = BE/SE, where BE is the total space of the n-ball bundleassociated with ξ and SE = ∂BE is the (n − 1)-sphere bundle. Also, Th ξ is theone-point compactification of E. The Thom space Th ξ has a canonical basepoint,the image of E>1.

Proposition D.2.1. If ξ and η are vector bundles over X and Y respectively,and ξ × η is the product vector bundle over X × Y , then

Th(ξ × η) = Th ξ ∧ Th η.

The proof is left as an exercise.

Example D.2.2.1. Regarding Rk as the total space of a k-plane bundle over a point, we obtain

that the corresponding Thom space Th(Rk) is a k-sphere Sk.2. If ξ is a 0-dimensional bundle over X , then Th ξ = X+ = X ⊔ pt.3. Let Rk denote the trivial k-plane bundle over X . The Whitney sum ξ ⊕ Rk

can be identified with the product bundle ξ × Rk, where Rk is the k-plane bundleover a point. Then Proposition D.2.1 implies that

Th(ξ ⊕ Rk) = Σk Th ξ,

where Σk denote the k-fold suspension.4. Combining the previous two examples, we obtain

Th(Rk) = ΣkX ∨ Sk.

Construction D.2.3 (Pontryagin–Thom construction). Let M be a subman-ifold in Rm with normal bundle ν = ν(M ⊂ Rm). The Pontryagin–Thom map

Sm → Th ν

identifies the tubular neighbourhood of M in Rm ⊂ Sm with the set of vectorsof length < 1 in the fibres of ν, and collapses the complement of the tubularneighbourhood to the basepoint of the Thom space Th ν.

D.2. THOM SPACES AND COBORDISM FUNCTORS 443

This construction can be generalised to submanifolds M ⊂ Eξ in the totalspace of an arbitrary m-plane bundle ξ over a manifold, giving the collapse map

(D.1) Th ξ → Th ν,

where ν = ν(M ⊂ Eξ). Note that the Pontryagin–Thom collapse map is a partic-ular case of (D.1), as Sm is the Thom space of an m-plane bundle over a point.

Recall that a smooth map f : W → Z of manifolds is called transverse along asubmanifold Y ⊂ Z if, for every w ∈ f−1(Y ), the image of the tangent space to Wat w together with the tangent space to Y at f(w) spans the tangent space to Zat f(w):

f∗TwW + Tf(w)Y = Tf(w)Z.

If f : W → Z is transverse along Y ⊂ Z, then f−1(Y ) is a submanifold in W ofcodimension equal to the codimension of Y in Z.

Construction D.2.4 (cobordism classes of η-submanifolds in Eξ). Let ξ bean m-plane bundle with total space Eξ over a manifold X , and let η be an n-planebundle over a manifold Y . An η-submanifold of Eξ is a pair (M, f) consisting of asubmanifold M in Eξ and a map

f : ν(M ⊂ Eξ)→ η

such that f is an isomorphism on each fibre (so that the codimension of M inEξ is n). Two η-submanifolds (M0, f0) and (M1, f1) are cobordant if there is anη-submanifold with boundary (W, f) in the cylinder Eξ × I ⊂ E(ξ ⊕ R) such that

∂(W, f) =((M0, f0)× 0

)∪((M1, f1)× 1

).

Theorem D.2.5. The set of cobordism classes of η-submanifolds in Eξ is inone-to-one correspondence with the set [Th ξ,Th η] of homotopy classes of basedmaps of Thom spaces.

Proof. Assume given a based map g : Th ξ → Th η. By changing g within itshomotopy class we may achieve that g is transverse along the zero section Y ⊂ Th η(transversality is a local condition, and both Th ξ and Th η are manifolds outsidethe basepoints). Since η is an n-plane bundle, M = g−1(Y ) is a submanifold ofcodimension n in Eξ = Th ξ \ pt such that

ν(M ⊂ Eξ) = g∗(ν(Y ⊂ Eη)) = g∗η.

That is, M is an η-submanifold in Eξ.Conversely, assume given an η-submanifold M ⊂ Eξ. We therefore have the

map of Thom spaces Th ν → Th η (induced by the map of ν to η), whose compo-sition with the collapse map (D.1) gives the required map Th ξ → Th η.

The fact that homotopic based maps Th ξ → Th η correspond to cobordantη-submanifolds, and vice versa, is left as an exercise.

Construction D.2.6 (cobordism groups). Let ηk be the universal vector k-plane bundle EO(k)→ BO(k). Following the original notation of Thom, we denoteMO(k) = Th ηk.

Every submanifold M ⊂ Rn+k of dimension n is an ηk-submanifold via the

classifying map of the normal bundle ν(M ⊂ Rn+k). Denote by Ω−n,kO the set of

cobordism classes of M ⊂ Rn+k. The base of ηk is not a manifold, but it is a direct


limit of Grassmannians, and a simple limit argument shows that Theorem D.2.5still holds for ηk-submanifolds. Hence,

Ω−n,kO = [Sn+k,Th ηk] = πn+k(MO(k)).

There is the stabilisation map Ω−n,kO → Ω−n,k+1

O obtained by composing the sus-pended map Sn+k+1 → ΣMO(k) with the map ΣMO(k) → MO(k + 1) inducedby the bundle map ηk ⊕ R→ ηk+1. The (−n)th cobordism group is defined by

(D.2) Ω−nO = limk→∞Ω−n,k

O = limk→∞ πn+k(MO(k)).

Proposition D.2.7. We have a canonical isomorphism

Ω−nO∼= ΩOn

between the cobordism and bordism groups, for n > 0. In other words, two n-dimensional manifolds M0 and M1 are bordant if and only if there exist embeddingsof M0 and M1 in the same Rn+k which are cobordant.

Proof. Forgetting the embedding M ⊂ Rn+k we obtain a map Ω−n,kO → ΩOn ,

which may be shown to be a group homomorphism. It is compatible with thestabilisation maps, and therefore defines a homomorphism Ω−n

O → ΩOn . Sinceevery manifold M may be embedded in some Rn+k, it is an isomorphism.

Together with (D.2), Proposition D.2.7 gives a homotopical interpretation forthe (unoriented) bordism groups. This also implies that the notions of the ‘bordismclass’ and ‘cobordism class’ of a manifold M are interchangeable. Theorem D.2.5may be applied further to obtain a homotopical interpretation for the bordismgroups On(X) of a space X :

Construction D.2.8 (bordism and cobordism groups of a space). Let X bea space. We set ξ = Rn+k (an (n + k)-plane bundle over a point) and η = X ×ηk (the product of a 0-plane bundle over X and the universal k-plane bundle ηkover BO(k)), and consider cobordism classes of η-submanifolds in Eξ. Such asubmanifold is described by a pair (f, ι) consisting of a map f : M → X from ann-dimensional manifold to X and an embedding ι : M → Rn+k (the bundle mapfrom ν(ι) to η is the product of the map Eν(ι)→M → X and the classifying mapof ν(ι)). By Theorem D.2.5, the set of cobordism classes of η-submanifolds in Eξis given by

[Sn+k,Th(X × ηk)] = πn+k((X+) ∧MO(k)

).

As in the proof of Proposition D.2.7, there is the map from the above set of cobor-dism classes to the bordism group On(X), which forgets the embedding M ⊂ Rn+k.A stabilisation argument shows that

(D.3) On(X) = limk→∞ πk+n((X+) ∧MO(k)

),

providing a homotopical interpretation for the bordism groups of X .We define the cobordism groups of X as

(D.4) On(X) = limk→∞

[Σk−n(X+),MO(k)

].

If X is a (not necessarily compact) manifold, then the groups On(X) may alsobe obtained by stabilising the set of cobordism classes of η-submanifolds in Eξ.Namely, we need to set ξ = Rk−n (the trivial (k − n)-plane bundle over X), andη = ηk. In other words, a cobordism class in On(X) is described by the composition

(D.5) M → X × Rk−n −→ X,

D.3. ORIENTED AND COMPLEX BORDISM 445

where the first map is an embedding of codimension k.The maps of Thom spaces MO(k) ∧ MO(l) → MO(k + l) (induced by the

classifying maps ηk × ηl → ηk+l) turn O∗(X) =∏n∈ZO

n(X) into a graded ring,called the unoriented cobordism ring of X . (One needs to consider direct productinstead of direct sum to take care of infinite complexes like RP∞.)

Exercises.

D.2.9. Prove Proposition D.2.1.

D.2.10. If η is the tautological line bundle over RPn (respectively, CPn), thenTh η can be identified with RPn+1 (respectively, CPn+1).

D.2.11. Prove that cobordism of η-submanifolds is an equivalence relation.

D.2.12. Complete the proof of Theorem D.2.5.

D.2.13. The forgetful map Ω−n,kO → ΩOn is a homomorphism of groups.

D.2.14. Given any (k−n)-plane bundle ξ over a manifold X and an embeddingM → Eξ of codimension k, the composition

M → Eξ −→ X

determines a cobordism class in On(X). (Hint: reduce to (D.5) by embedding ξinto a trivial bundle over the same X .)

D.2.15 (Poincare–Atiyah duality [7] in unoriented bordism). If X is an n-dimensional manifold, then

On−k(X) = Ok(X) for any k.

In particular, for X = pt we obtain the isomorphisms of Proposition D.2.7.

D.3. Oriented and complex bordism

The bordism relation may be extended to manifolds endowed with some addi-tional structure, which leads to important bordism theories.

The simplest additional structure is an orientation. By definition, two orientedn-dimensional manifolds M1 and M2 are oriented bordant if there is an oriented(n+ 1)-dimensional manifold W with boundary such that ∂W =M1 ⊔M2, whereM2 denotes M2 with the orientation reversed. The oriented bordism groups ΩSOnand the oriented bordism ring ΩSO =

⊕n>0Ω

SOn are defined accordingly. Given

an oriented manifold M , the manifold M × I has the canonical orientation suchthat ∂(M × I) = M ⊔M . Hence, −[M ] = [M ] in ΩSOn . Unlike ΩOn , elements ofΩSO generally do not have order 2.

Complex structure gives another important example of an additional structureon manifolds. However, a direct attempt to define the bordism relation on complexmanifolds fails because the manifold W must be odd-dimensional and thereforecannot be complex. This can be remedied by considering stably complex (alsoknown as stably almost complex or quasicomplex ) structures.

Let TM denote the tangent bundle of M . We say that M admits a tangentialstably complex structure if there is an isomorphism of real vector bundles

(D.6) cT : TM ⊕ Rk → ξ


between the ‘stable’ tangent bundle and a complex vector bundle ξ over M . Someof the choices of such isomorphisms are deemed to be equivalent, that is, determinethe same stably complex structures. This equivalence relation is generated by

(a) additions of trivial complex summands; that is, cT is equivalent to

TM ⊕ Rk ⊕ CcT ⊕id−−−→ ξ ⊕ C,

where C in the left hand side is canonically identified with R2;(b) compositions with isomorphisms of complex bundles; that is, cT is equiv-

alent to ϕ · cT for every C-linear isomorphism ϕ : ξ → ζ.

The equivalence class of cT may be described homotopically as the equivalence classof lifts of the map M → BO(2N) classifying the stable tangent bundle to a mapM → BU(N) up to homotopy and stabilisation (see [297, Chapters II, VII]). Atangential stably complex manifold is a pair consisting of M and an equivalenceclass of isomorphisms cT ; we shall use a simplified notation (M, cT ) for such pairs.This notion is a generalisation of complex and almost complex manifolds (wherethe latter means a manifold with a choice of a complex structure on TM , that is, astably complex structure (D.6) with k = 0).

We say that M admits a normal complex structure if there is an embeddingi : M → RN with the property that the normal bundle ν(i) admits a structureof a complex vector bundle. There is an appropriate notion of stable equivalencefor such embeddings i, and a normal complex structure cν on M is defined as thecorresponding equivalence class. Tangential and normal stably complex structuresonM determine each other by means of the canonical isomorphism TM⊕ν(i) ∼= RN .

Example D.3.1. Let M = CP 1. The standard complex structure on M isequivalent to the stably complex structure determined by the isomorphism

T (CP 1)⊕ R2 ∼=−→ η ⊕ ηwhere η is the tautological line bundle. On the other hand, one can view CP 1

as S2 embedded into R4 ∼= C2 with trivial normal bundle. We therefore have anisomorphism

T (CP 1)⊕ R2 ∼=−→ C2 ∼= η ⊕ ηwhich determines a trivial stably complex structure on CP 1.

The bordism relation can be defined between stably complex manifolds by tak-ing account of the stably complex structure in the bordism relation. As in the caseof unoriented bordism, the set of bordism classes [M, cT ] of n-dimensional stablycomplex manifolds is an Abelian group with respect to disjoint union. This groupis called the n-dimensional complex bordism group and denoted by ΩUn . The sphereSn has the canonical normally complex structure determined by a complex struc-ture on the trivial normal bundle of the embedding Sn → Rn+2. The correspondingbordism class represents the zero element in ΩUn . The opposite element to the bor-dism class [M, cT ] in the group ΩUn may be represented by the same manifold Mwith the stably complex structure determined by the isomorphism

TM ⊕ Rk ⊕ CcT ⊕τ−−−→ ξ ⊕ C

where τ : C→ C is the complex conjugation. We shall use the abbreviated notations[M ] and [M ] for the complex bordism class and its opposite whenever the stably


complex structure cT is clear from the context. There is a stably complex structureon M × I such that ∂(M × I) =M ⊔M .

The direct product of stably complex manifolds turns ΩU =⊕

n>0ΩUn into a

graded ring, called the complex bordism ring.

Construction D.3.2 (homotopic approach to cobordism). The complex bor-dism groups Un(X) and cobordism groups Un(X) may be defined homotopicallysimilarly to (D.3) and (D.4):

(D.7)Un(X) = limk→∞ π2k+n

((X+) ∧MU (k)

),

Un(X) = limk→∞

[Σ2k−n(X+),MU (k)

],

where MU (k) is the Thom space of the universal complex k-plane bundle EU (k)→BU (k). Here the direct limit uses the maps Σ2MU(k)→MU(k + 1).

Construction D.3.3 (geometric approach to cobordism). Both groups Un(X)and Un(X) may also be defined geometrically, in a way similar to the geometricconstruction of unoriented bordism and cobordism groups (Construction D.2.8).The complex bordism group Un(X) consists of bordism classes of maps M → X ofstably complex n-dimensional manifolds M to X .

The complex cobordism group Un(X) of a manifold X may be defined viacobordisms of η-submanifolds in Eξ, like in the unoriented case. Let ξ = X×R2k−n

(a trivial bundle), and let η be the canonical (universal) complex k-plane bundleover BU(k). Then an η-submanifold M in Eξ defines a composite map of manifolds

M → X × R2k−n −→ X

(compare (D.5)), where the first map is an embedding whose normal bundle has astructure of a complex k-plane bundle, and the second map is the projection ontothe first factor. A map M → X between manifolds which can be decomposed asabove is said to be complex orientable of codimension n. A choice of this decom-position together with a complex bundle structure in the normal bundle is called acomplex orientation of the map M → X . As usual, the equivalence relation on theset of complex orientations of M → X is generated by bundle isomorphisms andstabilisations. The group Un(X) consists of cobordism classes of complex orientedmaps M → X of codimension n.

Let y ∈ Un(Y ) be a cobordism class represented by a complex oriented mapM → Y , and let f : X → Y be a map of manifolds. If these two maps are transverse,the cobordism class f∗(y) ∈ Un(X) is represented by the pullback X ×Y M → Xwith the induced complex orientation.

When M → X is a fibre bundle with fibre F , the normal structure used inthe definition of a complex orientation can be converted to a tangential structure.Namely, an equivalence class of complex orientations of the bundle projection M →X is determined by a choice of stably complex structure for the bundle TF (M) oftangents along the fibres of M → X (an exercise). Such a bundle M → X is calledstably tangentially complex.

The equivalence of the homotopic and geometric approaches to cobordism isestablished using transversality arguments and the Pontryagin–Thom construction,as in the unoriented case.

If X = pt, then we obtain

U−n(pt) = Un(pt) = ΩUn


for n > 0, from either the homotopic or geometric description of the (co)bordismgroups. We also set Ω−n

U = U−n(pt) and ΩU =⊕

n>0Ω−nU .

Construction D.3.4 (pairing and products). The product operations incobordism are defined using the maps of Thom spaces MU (k)∧MU (l)→ MU (k+l)induced by the classifying maps of the products of canonical bundles.

There is a canonical pairing (the Kronecker product)

〈 , 〉 : Um(X)⊗ Un(X)→ ΩUn−m,

the -product

: Um(X)⊗ Un(X)→ Un−m(X),

and the -product (or simply product)

: Um(X)⊗ Un(X)→ Um+n(X),

defined as follows. Assume given a cobordism class x ∈ Um(X) represented by amap Σ2l−mX+ → MU(l) and a bordism class α ∈ Un(X) represented by a mapS2k+n → X+ ∧MU(k). Then 〈x, α〉 ∈ ΩUn−m is represented by the composite map

S2k+2l+n−m Σ2l−mα−−−−−→ Σ2l−mX+ ∧MU(k)x∧ id−−−−→ MU(l) ∧MU(k)→MU(l+k)

If ∆ : X+ → (X ×X)+ = X+ ∧X+ is the diagonal map, then x α ∈ Un−m(X)is represented by the composite map

S2k+2l+n−m Σ2l−mα−−−−−→ Σ2l−mX+ ∧MU(k)Σ2l−m∆∧ id−−−−−−−−→ X+ ∧Σ2l−mX+ ∧MU(k)

id∧x∧id−−−−−→ X+ ∧MU(l) ∧MU(k)→ X+ ∧MU(l+ k)

The -product is defined similarly; it turns U∗(X) =∏n∈Z U

n(X) into a gradedring, called the complex cobordism ring of X . It is a module over ΩU .

The products operations can be also interpreted geometrically.For example, assume that x ∈ Um(X) is represented by an embedding of man-

ifolds Mk−m → X = Xk with a complex structure in the normal bundle, andα ∈ Un(X) is represented by an embedding Nn → Xk of a tangentially stablycomplex manifold N . Assume further that M and N intersect transversely in X ,i.e. dimM ∩N = n−m. Then 〈x, α〉 is the bordism class of the intersection M ∩N ,and x α is the bordism class of the embedding M ∩ N → X . The tangentialcomplex structure of M ∩ N is defined by the tangential structure of N and thecomplex structure in the normal bundle of M ∩N → N induced from the normalbundle of M → X .

Similarly, if x ∈ U−d(X) is represented by a smooth fibre bundle Ek+d → Xk

and α ∈ Un(X) is represented by a smooth map N → X , then 〈x, α〉 ∈ ΩUn+d is thebordism class of the pull-back E′, and x α ∈ Un+d(X) is the bordism class ofthe composite map E′ → X in the pull-back diagram

E′ −−−−→ Ey

y

N −−−−→ X

Construction D.3.5 (Poincare–Atiyah duality in cobordism). Let X be amanifold of dimension d. The inclusion of a point pt ⊂ X defines the bordismclass 1 ∈ U0(X) and the fundamental cobordism class of X in Ud(X). (The normal


bundle of a point has a complex structure if d is even, otherwise the normal bundleof a point in X × R has a complex structure.)

The identity map X → X defines the cobordism class 1 ∈ U0(X). It definesthe fundamental bordism class of X in Ud(X) only when X is stably complex.

Now let X be stably complex manifold with fundamental bordism class [X ] ∈Ud(X). The map

D = · [X ] : Uk(X)→ Ud−k(X), x 7→ x [X ]

is an isomorphism (an exercise); it is called the Poincare–Atiyah duality map.

Construction D.3.6 (Gysin homomorphism). Let f : Xk → Y k+d be a com-plex oriented map of codimension d between manifolds (manifolds may be notcompact, in which case f is assumed to be proper). It induces a covariant map

f ! : Un(X)→ Un+d(Y )

called the Gysin homomorphism, whose geometric definition is as follows. Letx ∈ Un(X) be represented by a complex oriented map g : Mk−n → Xk. Then f !(x)is represented by the composition fg.

Proposition D.3.7. The Gysin homomorphism has the following properties:

(a) f ! : U∗(X)→ U∗+d(Y ) depends only on the homotopy class of f ;

(b) f ! is a homomorphism of ΩU -modules;(c) (fg) ! = f !g !;(d) f !(x · f∗(y)) = f !(x) · y for any x ∈ Un(X), y ∈ Um(Y );(e) assume that

X ×Y Z g′−−−−→ X

f ′

yyf

Zg−−−−→ Y

is a pullback square of manifolds, where g is transverse to f and f ′ isendowed with the pullback of the complex orientation of f . Then

g∗f ! = f ′! g

′∗ : U∗(X)→ U∗+d(Z).

Proof. (a) is clear from the homotopic definition of cobordism, while the otherproperties follow easily from the geometric definition. For example, to prove (d)one needs to choose maps Z → X and W → Y representing x and y, respectively,and consider the commutative diagram

Z ×Y W −−−−→ X ×Y W −−−−→ Wy

yy

Z −−−−→ Xf−−−−→ Y

Both sides of (d) are represented by the composite map Z ×Y W → Y above.

Let ξ be a complex n-plane bundle with total space E over a manifold X andlet i : X → E be the zero section. The element i∗i !1 ∈ U2n(X) where 1 ∈ U0(X)is called the Euler class of ξ and is denoted by e(ξ).

Construction D.3.8 (geometric cobordisms). For any cell complex X thecohomology group H2(X) can be identified with the set [X,CP∞] of homotopyclasses of maps into CP∞. Since CP∞ = MU(1), it follows from (D.7) that every


element x ∈ H2(X) determines a cobordism class ux ∈ U2(X). The elements ofU2(X) obtained in this way are called geometric cobordisms of X . We thereforemay view H2(X) as a subset in U2(X), however the group operation in H2(X) isnot obtained by restricting the group operation in U2(X) (the relationship betweenthe two operations is discussed in Appendix E).

WhenX is a manifold, geometric cobordisms may be described by submanifoldsM ⊂ X of codimension 2 with a fixed complex structure on the normal bundle.

Indeed, every x ∈ H2(X) corresponds to a homotopy class of maps fx : X →CP∞. The image fx(X) is contained in some CPN ⊂ CP∞, and we may assumethat fx(X) is transverse to a certain hyperplane H ⊂ CPN . Then Mx = f−1

x (H)is a codimension-2 submanifold in X whose normal bundle acquires a complexstructure by restriction of the complex structure on the normal bundle ofH ⊂ CPN .Changing the map fx within its homotopy class does not affect the bordism classof the embedding Mx → X .

Conversely, assume given a submanifoldM ⊂ X of codimension 2 whose normalbundle is endowed with a complex structure. Then the composition

X → Th(ν)→MU(1) = CP∞

of the Pontryagin–Thom collapse map X → Th(ν) and the map of Thom spacescorresponding to the classifying map M → BU(1) of ν defines an element xM ∈H2(X), and therefore a geometric cobordism.

If X is an oriented manifold, then a choice of complex structure on the normalbundle of a codimension-2 embedding M ⊂ X is equivalent to orienting M . Theimage of the fundamental class of M in H∗(X) is Poincare dual to xM ∈ H2(X).

Construction D.3.9 (connected sum). For manifolds of positive dimensionthe disjoint union M1 ⊔M2 representing the sum of bordism classes [M1] + [M2]may be replaced by their connected sum, which represents the same bordism class.

The connected sum M1 #M2 of manifolds M1 and M2 of the same dimensionn is constructed as follows. Choose points v1 ∈ M1 and v2 ∈ M2, and take closedε-balls Bε(v1) and Bε(v2) around them (both manifolds may be assumed to beendowed with a Riemannian metric). Fix an isometric embedding f of a pair ofstandard ε-balls Dn×S0 (here S0 = 0, 1) into M1 ⊔M2 which maps Dn× 0 ontoBε(v1) and Dn × 1 onto Bε(v2). If both M1 and M2 are oriented we additionallyrequire the embedding f to preserve the orientation on the first ball and reverse inon the second. Now, using this embedding, replace in M1 ⊔M2 the pair of ballsDn × S0 by a ‘pipe’ Sn−1 ×D1. After smoothing the angles in the standard waywe obtain a smooth manifold M1 #M2.

If both M1 and M2 are connected the smooth structure on M1 # M2 doesnot depend on a choice of points v1, v2 and embedding Dn × S0 → M1 ⊔M2. Itdoes however depend on the orientations; M1 #M2 and M1 #M2 are not diffeo-morphic in general. For example, the manifolds CP 2 # CP 2 and CP 2 # CP 2 arenot diffeomorphic (and not even homotopy equivalent because they have differentsignatures).

There are smooth contraction maps p1 : M1 #M2 →M1 and p2 : M1 #M2 →M2. In the oriented case the manifold M1#M2 can be oriented in such a way thatboth contraction maps preserve the orientations.

A bordism between M1 ⊔ M2 and M1 # M2 may be constructed as follows.Consider a cylinder M1× I, from which we remove an ε-neighbourhood Uε(v1 × 1)


Figure D.2. Disjoint union and connected sum.

of the point v1 × 1. Similarly, remove the neighbourhood Uε(v2 × 1) from M2 × I(each of these two neighbourhoods can be identified with the half of a standardopen (n + 1)-ball). Now connect the two remainders of cylinders by a ‘half pipe’Sn6 × I in such a way that the half-sphere Sn6 × 0 is identified with the half-sphere

on the boundary of Uε(v1× 1), and Sn6× 1 is identified with the half-sphere on the

boundary of Uε(v2×1). Smoothing the angles we obtain a manifold with boundaryM1⊔M2 ⊔ (M1#M2) (or M1⊔M2 ⊔ (M1#M2) in the oriented case), see Fig. D.2.

Finally, if M1 and M2 are stably complex manifolds, then there is a canonicalstably complex structure on M1#M2, which is constructed as follows. Assume thestably complex structures on M1 and M2 are determined by isomorphisms

cT,1 : TM1 ⊕ Rk1 → ξ1 and cT,2 : TM2 ⊕ Rk2 → ξ2.

Using the isomorphism T (M1 #M2) ⊕ Rn ∼= p∗1TM1 ⊕ p∗2TM2, we define a stablycomplex structure on M1 #M2 by the isomorphism

T (M1 #M2)⊕ Rn+k1+k2

∼= p∗1TM1 ⊕ Rk1 ⊕ p∗2TM2 ⊕ Rk2cT ,1⊕cT ,2−−−−−−→ p∗1ξ1 ⊕ p∗2ξ2.

We shall refer to this stably complex structure as the connected sum of stablycomplex structures on M1 and M2. The corresponding complex bordism class is[M1] + [M2].

Exercises.

D.3.10. Assume given a complex (k− l)-plane bundle ξ over a manifold X andan embedding M → Eξ whose normal bundle has a structure of a complex k-planebundle. Then the composition

M → Eξ −→ X

determines a complex orientation for the map M → X of codimension 2l, andtherefore a complex cobordism class in U2l(X). (Compare Exercise D.2.14.)

Similarly, an embedding M → E(ξ ⊕ R) whose normal bundle has a structureof a complex k-plane bundle determines a complex cobordism class in U2l−1(X),via the composition

M → E(ξ ⊕ R) −→ X.

This is how complex orientations were defined in [273].

D.3.11. Let π : E → B be a bundle with fibre F . The map π is complex orientedif and only a stably complex structure is chosen for the bundle TF (E) of tangentsalong the fibres of π.

D.3.12. The Poincare–Atiyah duality map

D = · [X ] : Uk(X)→ Ud−k(X), x 7→ x [X ]

is an isomorphism for any stably complex manifold X of dimension d.

D.3.13. Let f : Xd → Y p+d be a complex oriented map of manifolds, and letDX : Uk(X) → Ud−k(X), DY : Up+k(Y ) → Ud−k(Y ) be the duality isomorphismsfor X , Y . Then the Gysin homomorphism satisfies f ! = D−1

Y f∗DX .


D.3.14. Let ξ be a complex n-plane bundle over a manifold M with totalspace E, and let i : M → E be the inclusion of zero section. Define the Gysinhomomorphism

i ! : U∗(M)→ U∗+2n(E,E \M) = U∗+2n(Th(ξ))

by analogy with Construction D.3.6 and show that i ! is an isomorphism. It is calledthe Gysin–Thom isomorphism corresponding to ξ.

D.4. Structure results

Let M be an n-dimensional manifold and let f : M → BO(n) be the classify-ing map of the tangent bundle. Given a universal Stiefel–Whitney characteristicclass w ∈ H∗(BO(n);Z2) = Z2[w1, . . . , wn] of n-plane bundles, the correspond-ing corresponding tangential Stiefel–Whitney characteristic number w[M ] is de-fined as the result of pairing of f∗(w) ∈ Hn(M ;Z2) with the fundamental class〈M〉 ∈ Hn(M ;Z2). The number w[M ] is an unoriented bordism invariant.

Tangential Chern characteristic numbers c[M ] of stably complex 2n-manifoldsand Pontryagin characteristic numbers p[M ] of oriented 4n-manifolds are definedsimilarly; they are complex and oriented bordism invariants, respectively.

Normal characteristic numbers are of equal importance in cobordism theory. IfM → RN is an embedding with a fixed complex structure in the normal bundle ν,classified by the map g : ν → BU(n), then the normal Chern characteristic numberc[M ] corresponding to c ∈ H∗(BU(n)) = Z[c1, . . . , cn] is defined as (g∗c)〈M〉.Normal Stiefel–Whitney and Pontryagin numbers are defined similarly. Since TM⊕ν = RN , the tangential and normal characteristic numbers determine each other.

In what follows, all characteristic numbers will be tangential.

The theory of unoriented (co)bordism was the first to be completed: the co-efficient ring ΩO was calculated by Thom, and the bordism groups O∗(X) of cellcomplexes X were reduced to homology groups of X with coefficients in ΩO. Thecorresponding results are summarised as follows.

Theorem D.4.1.

(a) Two manifolds are unorientedly bordant if and only if they have identicalsets of Stiefel–Whitney characteristic numbers.

(b) ΩO is a polynomial ring over Z2 with one generator ai in every positivedimension i 6= 2k − 1.

(c) For every cell complex X the module O∗(X) is a free graded ΩO-moduleisomorphic to H∗(X ;Z2)⊗Z2 Ω

O.

Parts (a) and (b) were done by Thom [305]. Part (c) was first formulated byConner and Floyd [81]; it also follows from the results of Thom.

Theorem D.4.2.

(a) ΩU ⊗ Q is a polynomial ring over Q generated by the bordism classes ofcomplex projective spaces CP i, i > 1.

(b) Two stably complex manifolds are bordant if and only if they have identicalsets of Chern characteristic numbers.

(c) ΩU is a polynomial ring over Z with one generator ai in every even di-mension 2i, where i > 1.

D.5. RING GENERATORS 453

Part (a) can be proved by the methods of Thom. Part (b) follows from theresults of Milnor [226] and Novikov [244]. Part (c) is the most difficult one; it wasdone by Novikov [244] using the Adams spectral sequence and structure theoryof Hopf algebras (see also [245] for a more detailed account) and Milnor (unpub-lished1). Another more geometric proof was given by Stong [297].

Note that part (c) of Theorem D.4.1 does not extend to complex bordism;U∗(X) is not a free ΩU -module in general (although it is a free ΩU -module ifH∗(X ;Z) is free abelian). Unlike the case of unoriented bordism, the calculationof complex bordism of a space X does not reduce to calculating the coefficient ringΩU and homology groups H∗(X). The theory of complex (co)bordism is muchricher than its unoriented analogue, and at the same time is not as complicatedas oriented bordism or other bordism theories with additional structure, since thecoefficient ring does not have torsion. Thanks to this, complex cobordism theoryfound many striking and important applications in algebraic topology and beyond.Many of these applications were outlined in the pioneering work of Novikov [246].

The calculation of the oriented bordism ring was completed by Novikov [244](ring structure modulo torsion and odd torsion) and Wall [320] (even torsion), withimportant earlier contributions made by Rokhlin, Averbuch, and Milnor. Unlikecomplex bordism, the ring ΩSO has additive torsion. We give only a partial resulthere (which does not fully describe the torsion elements).

Theorem D.4.3.

(a) ΩSO ⊗Q is a polynomial ring over Q generated by the bordism classes ofcomplex projective spaces CP 2i, i > 1.

(b) The subring Tors ⊂ ΩSO of torsion elements contains only elements oforder 2. The quotient ΩSO/Tors is a polynomial ring over Z with onegenerator ai in every dimension 4i, where i > 1.

(c) Two oriented manifolds are bordant if and only if they have identical setsof Pontryagin and Stiefel–Whitney characteristic numbers.

D.5. Ring generators

To describe a set of ring generators for ΩU we shall need a special character-istic class of complex vector bundles. Let ξ be a complex k-plane bundle over amanifold M . Write its total Chern class formally as follows:

c(ξ) = 1 + c1(ξ) + · · ·+ ck(ξ) = (1 + x1) · · · (1 + xk),

so that ci(ξ) = σi(x1, . . . , xk) is the ith elementary symmetric function in formalindeterminates. These indeterminates acquire a geometric meaning if ξ is a sumξ1⊕ · · · ⊕ ξk of line bundles; then xj = c1(ξj), 1 6 j 6 k. Consider the polynomial

Pn(x1, . . . xk) = xn1 + · · ·+ xnk

and express it via the elementary symmetric functions:

Pn(x1, . . . , xk) = sn(σ1, . . . , σk).

Substituting the Chern classes for the elementary symmetric functions we obtain acertain characteristic class of ξ:

sn(ξ) = sn(c1(ξ), . . . , ck(ξ)) ∈ H2n(M).

1Milnor’s proof was announced in [159]; it was intended to be included in the second partof [226], but has never been published.


This characteristic class plays an important role in detecting the polynomial gener-ators of the complex bordism ring, because of the following properties (which followimmediately from the definition).

Proposition D.5.1. The characteristic class sn satisfies

(a) sn(ξ) = 0 if ξ is a bundle over M and dimM < 2n;(b) sn(ξ ⊕ η) = sn(ξ) + sn(η);(c) sn(ξ) = c1(ξ)

n if ξ is a line bundle.

Given a stably complex 2n-manifold (M, cT ), define its characteristic number

(D.8) sn[M ] = sn(TM)〈M〉 ∈ Z.

Here sn(TM) is understood to be sn(ξ), where ξ is the complex bundle from (D.6).

Corollary D.5.2. If a bordism class [M ] ∈ ΩU2n decomposes as [M1] × [M2]where dimM1 > 0 and dimM2 > 0, then sn[M ] = 0.

It follows that the characteristic number sn vanishes on decomposable elementsof ΩU2n. It also detects indecomposables that may be chosen as polynomial genera-tors. The following result featured in the proof of Theorem D.4.2.

Theorem D.5.3. A bordism class [M ] ∈ ΩU2n may be chosen as a polynomialgenerator an of the ring ΩU if and only if

sn[M ] =

±1, if n 6= pk − 1 for any prime p;

±p, if n = pk − 1 for some prime p.

There is no universal description of connected manifolds representing the poly-nomial generators an ∈ ΩU . On the other hand, there is a particularly nice familyof manifolds whose bordism classes generate the whole ring ΩU . This family isredundant though, so there are algebraic relations between their bordism classes.

Construction D.5.4 (Milnor hypersurfaces). Fix a pair of integers j > i > 0and consider the product CP i × CP j . Its algebraic subvariety

(D.9) Hij = (z0 : · · · : zi)× (w0 : · · · : wj) ∈ CP i × CP j : z0w0 + · · ·+ ziwi = 0is called a Milnor hypersurface. Note that H0j

∼= CP j−1.

Denote by p1 and p2 the projections of CP i × CP j onto its factors. Let η bethe tautological line bundle over a complex projective space and η its conjugate(the canonical line bundle). We have

H∗(CP i × CP j) = Z[x, y]/(xi+1 = 0, yj+1 = 0)

where x = p∗1c1(η), y = p∗2c1(η).

Proposition D.5.5. The geometric cobordism in CP i×CP j corresponding tothe element x + y ∈ H2(CP i × CP j) is represented by the submanifold Hij . Inparticular, the image of the fundamental class 〈Hij〉 in H2(i+j−1)(CP

i × CP j) isPoincare dual to x+ y.

Proof. We have x+ y = c1(p∗1(η)⊗ p∗2(η)). The classifying map fx+y : CP i ×

CP j → CP∞ is the composition of the Segre embedding

σ : CP i × CP j → CP (i+1)(j+1)−1,

(z0 : · · · : zi)× (w0 : · · · : wj) 7→ (z0w0 : z0w1 : · · · : zkwl : · · · : ziwj),

D.5. RING GENERATORS 455

and the embedding CP ij+i+j → CP∞. The codimension 2 submanifold in CP i ×CP j corresponding to the cohomology class x + y is obtained as the preimageσ−1(H) of a generally positioned hyperplane in CP ij+i+j (that is, a hyperplane Htransverse to the image of the Segre embedding, see Construction D.3.8). By (D.9),the Milnor hypersurface is exactly σ−1(H) for one such hyperplane H .

Lemma D.5.6.

si+j−1[Hij ] =

j, if i = 0;

2, if i = j = 1;

0, if i = 1, j > 1;

−(i+ji

), if i > 1.

Proof. Let i = 0. Since the stably complex structure on H0j = CP j−1 isdetermined by the isomorphism T (CP j−1) ⊕ C ∼= η ⊕ · · · ⊕ η (j summands) andx = c1(η), we have

sj−1[CPj−1] = jxj−1〈CP j−1〉 = j.

Now let i > 0. Then

si+j−1(T (CP i × CP j)) = (i+ 1)xi+j−1 + (j + 1)yi+j−1

=

2xj + (j + 1)yj, if i = 1;

0, if i > 1.

Denote by ν the normal bundle of the embedding ι : Hij → CP i × CP j . Then

(D.10) T (Hij)⊕ ν = ι∗(T (CP i × CP j)).

Since c1(ν) = ι∗(x+ y), we obtain si+j−1(ν) = ι∗(x + y)i+j−1.Assume i = 1. Then (D.10) and Proposition D.5.1 imply that

sj [H1j ] = sj(T (H1j)

)〈H1j〉 = ι∗

(2xj + (j + 1)yj − (x + y)j

)〈H1j〉

= (2xj + (j + 1)yj − (x+ y)j)(x + y)〈CP 1 × CP j〉 =2, if j = 1;

0, if j > 1.

Assume now that i > 1. Then si+j−1(T (CP i × CP j)) = 0, and we obtainfrom (D.10) and Proposition D.5.1 that

si+j−1[Hij ] = −si+j−1(ν)〈Hij〉 = −ι∗(x+ y)i+j−1〈Hij〉= −(x+ y)i+j〈CP i × CP j〉 = −

(i+ji

).

Remark. Since s1[H11] = 2 = s1[CP1] the manifold H11 is bordant to CP 1.

In fact H11∼= CP 1 (an exercise).

Theorem D.5.7. The bordism classes [Hij ], 0 6 i 6 j multiplicatively gen-erate the ring ΩU .

Proof. A simple calculation shows that

g.c.d.((n+1i

), 1 6 i 6 n

)=

p, if n = pk − 1,

1, otherwise.


Now Lemma D.5.6 implies that a certain integer linear combination of bordismclasses [Hij ] with i+ j = n+1 can be taken as the polynomial generator an of ΩU ,see Theorem D.5.3.

Remark. There is no universal description for a linear combination of bordismclasses [Hij ] with i+ j = n+ 1 giving the polynomial generator of ΩU .

All algebraic relations between the classes [Hij ] arise from the associativity ofthe formal group law of geometric cobordism (see Corollary E.2.4).

Example D.5.8. Since s1[CP 1] = 2, s2[CP 2] = 3, the bordism classes [CP 1]and [CP 2] may be taken as polynomial generators a1 and a2 of ΩU . However [CP 3]cannot be taken as a3, since s3[CP 3] = 4, while s3(a3) = ±2. The bordism class[H22] + [CP 3] may be taken as a3.

Theorem D.5.7 admits the following important addendum, which is due toMilnor (see [297, Chapter 7] for the proof).

Theorem D.5.9 (Milnor). Every bordism class x ∈ ΩUn with n > 0 contains anonsingular algebraic variety (not necessarily connected).

The proof of this fact uses a construction of a (possibly disconnected) alge-braic variety representing the class −[M ] for any bordism class [M ] ∈ ΩUn of 2n-dimensional manifold. The following question is still open.

Problem D.5.10 (Hirzebruch). Describe the set of bordism classes in ΩU con-taining connected nonsingular algebraic varieties.

Example D.5.11. The group ΩU2 is isomorphic to Z and is generated by [CP 1].Every class k[CP 1] ∈ ΩU2 contains a nonsingular algebraic variety, namely, a disjointunion of k copies of CP 1 for k > 0 and a Riemann surface of genus (1−k) for k 6 0.Connected algebraic varieties are contained only in the classes k[CP 1] with k 6 1.

Exercises.

D.5.12. Properties listed in Proposition D.5.1 determine the characteristic classsn uniquely.

D.5.13. Show that H11∼= CP 1.

D.5.14. Show that H1j is complex bordant to CP 1 × CP j−1. (Hint: calculatethe characteristic numbers; no geometric construction of this bordism is known!)

D.5.15. An alternative set of ring generators of ΩU can be constructed asfollows. Let M2n

k be a submanifold in CPn+1 dual to kx ∈ H2(CPn+1), where k isa positive integer, and x is the first Chern class of the hyperplane section bundle.For example, one can take M2n

k to be a nonsingular hypersurface of degree k. Thenthe set [M2n

k ] : n > 1, k > 1 multiplicatively generates the ring ΩU .

D.6. Invariant stably complex structures

Let M be a 2n-dimensional manifold with a stably complex structure deter-mined by the isomorphism

(D.11) cT : TM ⊕ R2(l−n) → ξ.

Assume that the torus T k acts on M .

D.6. INVARIANT STABLY COMPLEX STRUCTURES 457

Definition D.6.1. A stably complex structure cT is T k-invariant if for everyt ∈ T k the composition

(D.12) r(t) : ξc−1T−−→ TM ⊕ R2(l−n) dt ⊕ id−−−→ TM ⊕ R2(l−n) cT−→ ξ

is a complex bundle map, where dt is the differential of the action by t . In otherwords, (D.12) determines a representation r : T k → HomC(ξ, ξ).

Let x ∈ M be an isolated fixed point of the T k-action on M . Then we havea representation rx : T

k → GL(l,C) in the fibre of ξ over x. This fibre ξx ∼=Cl decomposes as Cn ⊕ Cl−n, where rx has no trivial summands on Cn and istrivial on Cl−n. The nontrivial part of rx decomposes into a sum ri1 ⊕ · · · ⊕ rinof one-dimensional complex T k-representations. In the corresponding coordinates(z1, . . . , zn), an element t = (e2πiϕ1 , . . . , e2πiϕk) ∈ T k acts by

t ·(z1, . . . , zn) = (e2πi〈w1, ϕ〉z1, . . . , e2πi〈wn, ϕ〉zn),

where ϕ = (ϕ1, . . . , ϕk) ∈ Rk and w j ∈ Zk, 1 6 j 6 n, are the weights of therepresentation rx at the fixed point x. Also, the isomorphism cT ,x of (D.11) inducesan orientation of the tangent space Tx(M).

Definition D.6.2. For any fixed point x ∈ M , the sign σ(x) is +1 if theisomorphism

Tx(M)id⊕ 0−−−→ Tx(M)⊕ R2(l−n) cT ,x−−→ ξx ∼= Cn ⊕ Cl−n

p−→ Cn,

respects the canonical orientations, and −1 if it does not; here p is the projectiononto the first summand.

So σ(x) compares the orientations induced by rx and cT ,x on Tx(M). If M isan almost complex T k-manifold (i.e. l = n) then σ(x) = 1 for every fixed point x.

Example D.6.3. We let the circle T 1 = S1 act on CP 1 in homogeneous co-ordinates by t · (z0 : z1) = (z0 : tz1). This action has two fixed points (0 : 1) and(1 : 0). In the standard stably complex structure (see Example D.3.1) the signs ofboth vertices are positive (this structure is complex and the map in Definition D.6.2is a complex linear map C→ C).

On the other hand, the trivial stably complex structure on CP 1 ∼= S2 may bethought of as induced from the embedding of S2 into R3 ⊕ R1 with trivial normalbundle, which can be made complex by identifying it with C. We therefore maythink of the circle action as the rotation of the unit sphere in R3 around the verticalaxis. The fixed points are the north and south poles. The rotations induced onthe tangent planes to the two fixed points are in different directions. Therefore thesigns of the two fixed points are different (which sign is positive depends on theconvention one uses to induce an orientation on S2 from the orientation of R3).

APPENDIX E

Formal group laws and Hirzebruch genera

The theory of formal groups originally appeared in algebraic geometry and playsan important role in number theory and cryptography. Formal groups laws werebrought into bordism theory in the pioneering work of Novikov [246], and provideda very powerful tool for the theory of group actions on manifolds and generalisedhomology theories. Early applications of formal group laws in cobordism concernedfinite group actions on manifolds, or ‘differentiable periodic maps’. Subsequentdevelopments included constructions of complex oriented cohomology theories andapplications to Hirzebruch genera, one of the most important class of invariants ofmanifolds.

E.1. Elements of the theory of formal group laws

Let R be a commutative ring with unit.A formal power series F (u, v) ∈ R[[u, v]] is called a (commutative one-

dimensional) formal group law over R if it satisfies the following equations:

(a) F (u, 0) = u, F (0, v) = v;(b) F (F (u, v), w) = F (u, F (v, w));(c) F (u, v) = F (v, u).

The original example of a formal group law over a field k is provided by theexpansion near the unit of the multiplication map G×G→ G in a one-dimensionalalgebraic group over k. This also explains the terminology.

A formal group law F over R is called linearisable if there exists a coordinatechange u 7→ gF (u) = u+

∑i>1 giu

i ∈ R[[u]] such that

(E.1) gF (F (u, v)) = gF (u) + gF (v).

Note that every formal group law over R determines a formal group law over R⊗Q.

Theorem E.1.1. Every formal group law F is linearisable over R⊗Q.

Proof. Consider the series ω(u) = ∂F (u,w)∂w

∣∣∣w=0

. Then

ω(F (u, v)) =∂F (F (u, v), w)

∂w

∣∣∣w=0

=∂F (F (u,w), v)

∂F (u,w)·∂F (u,w)

∂w

∣∣∣w=0

=∂F (u, v)

∂uω(u).

We therefore have duω(u) = dF (u,v)

ω(F (u,v)) . Set

(E.2) g(u) =

∫ u

0

dv

ω(v);

then dg(u) = dg(F (u, v)). This implies that g(F (u, v)) = g(u)+C. Since F (0, v) =v and g(0) = 0, we get C = g(v). Thus, g(F (u, v)) = g(u) + g(v).

459

460 E. FORMAL GROUP LAWS AND HIRZEBRUCH GENERA

A series gF (u) = u+∑i>1 giu

i satisfying equation (E.1) is called a logarithm ofthe formal group law F ; Theorem E.1.1 shows that a formal group law over R⊗Qalways has a logarithm. Its functional inverse series fF (t) ∈ R⊗Q[[t]] is called anexponential of the formal group law, so that we have F (u, v) = fF (gF (u) + gF (v))over R ⊗ Q. If R does not have torsion (i.e. R → R ⊗ Q is monomorphic), thelatter formula shows that a formal group law (as a series with coefficients in R) isfully determined by its logarithm (which is a series with coefficients in R⊗Q).

Example E.1.2. An example of a formal group law is given by the series

(E.3) F (u, v) = (1 + u)(1 + v)− 1 = u+ v + uv,

over Z, called the multiplicative formal group law. Introducing a formal indetermi-nate β of degree −2, we may consider the 1-parameter extension of the multiplica-tive formal group law, given by Fβ(u, v) = u + v − βuv, with coefficients in Z[β].Its logarithm and exponential series are given by

g(u) = − ln(1− βu)β

, f(x) =1− e−βx

β.

Let F =∑k,l aklu

kvl be a formal group law over a ring R and r : R → R′ a

ring homomorphism. Denote by r(F ) the formal series∑

k,l r(akl)ukvl ∈ R′[[u, v]];

then r(F ) is a formal group law over R′.A formal group law F over a ring A is universal if for any formal group law F

over any ring R there exists a unique homomorphism r : A→ R such that F = r(F).

Proposition E.1.3. If a universal formal group law F over A exists, then

(a) the ring A is multiplicatively generated by the coefficients of the series F ;(b) F is unique: if F ′ is another universal formal group law over A′, then

there is an isomorphism r : A→ A′ such that F ′ = r(F).

Proof. To prove the first statement, denote by A′ the subring in A generatedby the coefficients of F . Then there is a monomorphism i : A′ → A satisfying i(F) =F . On the other hand, by universality there exists a homomorphism r : A → A′

satisfying r(F) = F . It follows that ir(F) = F . This implies that ir = id: A→ Aby the uniqueness requirement in the definition of F . Thus A′ = A. The secondstatement is proved similarly.

Theorem E.1.4 (Lazard [192]). The universal formal group law F exists, andits coefficient ring A is isomorphic to the polynomial ring Z[a1, a2, . . .] on an infinitenumber of generators.

E.2. Formal group law of geometric cobordisms

The applications of formal group laws in cobordism theory build upon thefollowing fundamental construction.

Construction E.2.1 (Formal group law of geometric cobordisms [246]). LetX be a cell complex and u, v ∈ U2(X) two geometric cobordisms (see Construc-tion D.3.8) corresponding to elements x, y ∈ H2(X) respectively. Denote by u+

Hv

the geometric cobordism corresponding to the cohomology class x+ y.

E.2. FORMAL GROUP LAW OF GEOMETRIC COBORDISMS 461

Proposition E.2.2. The following relation holds in U2(X):

(E.4) u+Hv = FU (u, v) = u+ v +

∑

k>1, l>1

αkl ukvl,

where the coefficients αkl ∈ Ω−2(k+l−1)U do not depend on X. The series FU (u, v)

given by (E.4) is a formal group law over the complex cobordism ring ΩU .

Proof. We first do calculations with the universal example X = CP∞×CP∞.Then

U∗(CP∞ × CP∞) = Ω∗U [[u, v]],

where u, v are canonical geometric cobordisms given by the projections of CP∞ ×CP∞ onto its factors. We therefore have the following relation in U2(CP∞×CP∞):

(E.5) u+Hv =

∑

k,l>0

αkl ukvl,

where αkl ∈ Ω−2(k+l−1)U .

Now let the geometric cobordisms u, v ∈ U2(X) be given by maps fu, fv : X →CP∞ respectively. Then u = (fu × fv)

∗(u), v = (fu × fv)∗(v) and u +

Hv =

(fu × fv)∗(u +Hv), where fu × fv : X → CP∞ × CP∞. Applying the ΩU -module

map (fu × fv)∗ to (E.5) we obtain the required formula (E.4). The identities (a)and (c) for FU (u, v) are obvious, and the associativity (b) follows from the identity(u+

Hv) +

Hw = FU (FU (u, v), w) and the associativity of +

H.

Series (E.4) is called the formal group law of geometric cobordisms; nowadaysit is also usually referred to as the complex cobordism formal group law.

By definition, the geometric cobordism u ∈ U2(X) is the first Conner–FloydChern class cU1 (ξ) of the complex line bundle ξ over X obtained by pulling backthe canonical bundle along the map fu : X → CP∞ (it also coincides with theEuler class e(ξ) as defined in Section D.3). It follows that the formal group lawof geometric cobordisms gives an expression of cU1 (ξ ⊗ η) ∈ U2(X) in terms of theclasses u = cU1 (ξ) and v = cU1 (η) of the factors:

cU1 (ξ ⊗ η) = FU (u, v).

The coefficients of the formal group law of geometric cobordisms and its loga-rithms may be described geometrically by the following results.

Theorem E.2.3 (Buchstaber [47, Theorem 4.8]).

FU (u, v) =

∑i,j>0[Hij ]u

ivj(∑r>0[CP

r]ur)(∑

s>0[CPs]vs

) ,

where Hij (0 6 i 6 j) are Milnor hypersurfaces (D.9) and Hji = Hij.

Proof. Set X = CP i × CP j in Proposition E.2.2. Consider the Poincare–Atiyah duality map D : U2(CP i × CP j) → U2(i+j)−2(CP

i × CP j) (Construc-

tion D.3.5) and the map ε : U∗(CP i × CP j) → U∗(pt) = ΩU induced by the pro-jection CP i × CP j → pt. Then the composition

εD : U2(CP i × CP j)→ ΩU2(i+j)−2


takes geometric cobordisms to the bordism classes of the corresponding submani-folds. In particular, εD(u+

Hv) = [Hij ], εD(ukvl) = [CP i−k][CP j−l]. Applying εD

to (E.4) we obtain

[Hij ] =∑

k, l

αkl[CPi−k][CP j−l].

Therefore,∑

i,j

[Hij ]uivj =

(∑

k, l

αklukvl)(∑

i>k

[CP i−k]ui−k)(∑

j>l

[CP j−l]vj−l),

which implies the required formula.

Corollary E.2.4. The coefficients of the formal group law of geometric cobor-disms generate the complex cobordism ring ΩU .

Proof. By Theorem D.5.7, ΩU is generated by the cobordism classes [Hij ],which can be integrally expressed via the coefficients of FU using Theorem E.2.3.

Theorem E.2.5 (Mishchenko [246, Appendix 1]). The logarithm of the formalgroup law of geometric cobordisms is given by

gU (u) = u+∑

k>1

[CP k]

k + 1uk+1 ∈ ΩU ⊗Q[[u]].

Proof. By (E.2),

dgU (u) =du

∂FU (u,v)∂v

∣∣∣v=0

.

Using the formula of Theorem E.2.3 and the identity Hi0 = CP i−1, we calculate

dgU (u)

du=

1 +∑k>0[CP

k]uk

1 +∑

i>0([Hi1]− [CP 1][CP i−1])ui.

Now [Hi1] = [CP 1][CP i−1] (see Exercise D.5.14). It follows that dgU (u)du = 1 +∑

k>0[CPk]uk, which implies the required formula.

Using these calculations the following most important property of the formalgroup law FU can be easily established:

Theorem E.2.6 (Quillen [272, Theorem 2]). The formal group law FU ofgeometric cobordisms is universal.

Proof. Let F be the universal formal group law over a ring A. Then there is ahomomorphism r : A→ ΩU which takes F to FU . The series F , viewed as a formalgroup law over the ring A ⊗ Q, has the universality property for all formal grouplaws over Q-algebras. By Theorem E.1.1, such a formal group law is determined byits logarithm, which is a series with leading term u. It follows that if we write the

logarithm of F as∑bkuk+1

k+1 then the ring A⊗Q is the polynomial ring Q[b1, b2, . . .].

By Theorem E.2.5, r(bk) = [CP k] ∈ ΩU . Since ΩU ⊗Q ∼= Q[[CP 1], [CP 2], . . .], thisimplies that r ⊗Q is an isomorphism.

By Theorem E.1.4 the ring A does not have torsion, so r is a monomorphism.On the other hand, Theorem E.2.3 implies that the image r(A) contains the bordismclasses [Hij ] ∈ ΩU , 0 6 i 6 j. Since these classes generate the whole ring ΩU(Theorem D.5.7), the map r is onto and thus an isomorphism.

E.3. HIRZEBRUCH GENERA 463

Remark. Complex cobordism theory is the universal complex-oriented coho-mology theory, i.e. it is universal in the category of cohomology theories in which thefirst Chern class is defined for complex bundles. This implies that the formal grouplaw FU of geometric cobordisms is universal among formal group laws realised bycomplex-oriented cohomology theories. Theorem E.2.6 establishes the universalityof FU among all formal group laws. It therefore opens a way for application ofresults from the algebraic theory of formal groups in cobordism.

E.3. Hirzebruch genera

Every homomorphism ϕ : ΩU → R from the complex bordism ring to a com-mutative ring R with unit can be regarded as a multiplicative characteristic ofmanifolds which is an invariant of bordism classes. Such a homomorphism is calleda (complex) R-genus. (The term ‘multiplicative genus’ is also used, to emphasisethat such a genus is a ring homomorphism.)

Assume that the ring R does not have additive torsion. Then every R-genusϕ is fully determined by the corresponding homomorphism ΩU ⊗ Q → R ⊗ Q,which we shall also denote by ϕ. A construction due to Hirzebruch [160] describeshomomorphisms ϕ : ΩU ⊗Q→ R⊗Q by means of universal R-valued characteristicclasses of special type.

Construction E.3.1 (Hirzebruch genera). Let BU = limn→∞

BU(n). Then

H∗(BU) is isomorphic to the ring of formal power series Z[[c1, c2, . . .]] in universalChern classes, deg ck = 2k. The set of tangential Chern characteristic numbers ofa given manifold M defines an element in Hom(H∗(BU),Z), which belongs to thesubgroup H∗(BU) in the latter group. We therefore obtain a group homomorphism

(E.6) ΩU → H∗(BU).

Since the product in H∗(BU) arises from the maps BU(k) × BU(l) → BU(k + l)corresponding to the product of vector bundles, and the Chern classes have theappropriate multiplicative property, the map (E.6) is a ring homomorphism.

Part 2 of Theorem D.4.2 says that (E.6) is a monomorphism, and Part 1 ofthe same theorem says that the map ΩU ⊗Q→ H∗(BU ;Q) is an isomorphism. Itfollows that every homomorphism ϕ : ΩU ⊗ Q → R ⊗ Q can be interpreted as anelement of

HomQ(H∗(BU ;Q), R⊗Q) = H∗(BU ;Q)⊗R,or as a sequence of homogeneous polynomials Ki(c1, . . . , ci), i > 0, degKi = 2i.This sequence of polynomials cannot be chosen arbitrarily; the fact that ϕ is a ringhomomorphism imposes certain conditions. These conditions may be described asfollows: an identity

1 + c1 + c2 + · · · = (1 + c′1 + c′2 + · · · ) · (1 + c′′1 + c′′2 + · · · )implies the identity

(E.7)∑

n>0

Kn(c1, . . . , cn) =∑

i>0

Ki(c′1, . . . , c

′i) ·∑

j>0

Kj(c′′1 , . . . , c

′′j ).

A sequence of homogeneous polynomials K = Ki(c1, . . . , ci), i > 0 with K0 = 1satisfying the identities (E.7) is called a multiplicative Hirzebruch sequence.


Proposition E.3.2. A multiplicative sequence K is completely determined bythe series

Q(x) = 1 + q1x+ q2x2 + · · · ∈ R ⊗Q[[x]],

where x = c1, and qi = Ki(1, 0, . . . , 0); moreover, every series Q(x) as above deter-mines a multiplicative sequence.

Proof. Indeed, by considering the identity

(E.8) 1 + c1 + · · ·+ cn = (1 + x1) · · · (1 + xn)

we obtain from (E.7) that

Q(x1) · · ·Q(xn) = 1 +K1(c1) +K2(c1, c2) + · · ·+Kn(c1, . . . , cn) +Kn+1(c1, . . . , cn, 0) + · · · .

Along with the series Q(x) it is convenient to consider the series f(x) ∈ R ⊗Q[[x]] with leading term x given by the identity

Q(x) =x

f(x).

It follows that the nth term Kn(c1, . . . , cn) in the multiplicative Hirzebruch se-quence corresponding to a genus ϕ : ΩU ⊗ Q → R ⊗ Q is the degree-2n part ofthe series

∏ni=1

xi

f(xi)∈ R ⊗ Q[[c1, . . . , cn]]. By some abuse of notation, we regard∏n

i=1xi

f(xi)as a characteristic class of complex n-plane bundles. Then the value of

ϕ on an 2n-dimensional stably complex manifold M is given by

(E.9) ϕ[M ] =( n∏

i=1

xif(xi)

(TM))〈M〉,

where the Chern classes c1, . . . , cn are expressed via the indeterminates x1, . . . , xnby the relation (E.8), and the degree-2n part of

∏ni=1

xi

f(xi)is taken to obtain a

characteristic class of TM . We shall also denote the ‘characteristic class’∏ni=1

xi

f(xi)

of a complex n-plane bundle ξ by ϕ(ξ); so that ϕ[M ] = ϕ(TM)〈M〉.We refer to the homomorphism ϕ : ΩU → R⊗Q given by (E.9) as the Hirzebruch

genus corresponding to the series f(x) = x+ · · · ∈ R⊗Q[[x]].

Remark. A parallel theory of genera exists for oriented manifolds. Thesegenera are homomorphisms ΩSO → R from the oriented bordism ring, and theHirzebruch construction expresses genera over torsion-free rings via Pontryagincharacteristic classes (which replace the Chern classes).

Given a torsion-free ring R and a series f(x) ∈ R⊗Q[[x]] one may ask whetherthe corresponding Hirzebruch genus ϕ : ΩU → R ⊗ Q actually takes values in R.This constitutes the integrality problem for ϕ. A solution to this problem can beobtained using the theory of formal group laws.

Every genus ϕ : ΩU → R gives rise to a formal group law ϕ(FU ) over R, whereFU is the formal group of geometric cobordisms (Construction E.2.1).

Theorem E.3.3 (Novikov [247]). For every genus ϕ : ΩU → R, the exponentialof the formal group law ϕ(FU ) is the series f(x) ∈ R⊗Q[[x]] corresponding to ϕ.

This can be proved either directly, by appealing to the construction of geometriccobordisms, or indirectly, by calculating the values of ϕ on projective spaces andcomparing to the formula for the logarithm of the formal group law.


1st proof. Let X be a stably complex d-manifold with x, y ∈ H2(X), and letu, v ∈ U2(X) be the corresponding geometric cobordisms (see Construction D.3.8)represented by codimension-2 submanifolds Mx ⊂ X and My ⊂ X respectively.

Then ukvl ∈ U2(k+l)(X) is represented by a submanifold of codimension 2(k + l),which we denote by Mkl. By (E.4), we have the following relation in U2(X):

[Mx+y] =∑

k,l>0

αkl[Mkl]

where Mx+y ⊂ X is the codimension-2 submanifold dual to x+y ∈ H2(X). We ap-ply the composition εD of the Poincare–Atiyah duality map D : U2(X)→ Ud−2(X)and the augmentation Ud−2(X)→ ΩUd−2 to the identity above, and then apply the

genus ϕ to the resulting idenity in ΩUd−2 to obtain

(E.10) ϕ[Mx+y] =∑

ϕ(αkl)ϕ[Mkl].

Let ι : Mx+y ⊂ X be the embedding. Considering the decomposition

ι∗(T X) = TMx+y ⊕ ν(ι)and using the multiplicativity of the characteristic class ϕ we obtain

ι∗ϕ(T X) = ϕ(TMx+y) · ι∗( x+yf(x+y)).

Therefore,

(E.11) ϕ[Mx+y] = ι∗(ϕ(T X) · f(x+y)x+y

)〈Mx+y〉 =

(ϕ(T X) · f(x+ y)

)〈X〉.

Similarly, by considering the embedding Mkl → X we obtain

(E.12) ϕ[Mkl] =(ϕ(T X) · f(x)kf(y)l

)〈X〉.

Plugging (E.11) and (E.12) into (E.10) we finally obtain

f(x+ y) =∑

k,l>0

ϕ(αkl)f(x)kf(y)l.

This implies, by definition, that f is the exponential of ϕ(FU ).

2nd proof. The complex bundle isomorphism T (CP k) ⊕ C = η ⊕ · · · ⊕ η(k+1 summands) allows us to calculate the value of a genus on CP k explicitly. Letx = c1(η) ∈ H2(CP k) and let g be the series functionally inverse to f ; then

ϕ[CP k] =( x

f(x)

)k+1

〈CP k〉

= coefficient of xk in( x

f(x)

)k+1

= res0

( 1

f(x)

)k+1

=1

2πi

∮ ( 1

f(x)

)k+1

dx =1

2πi

∮1

uk+1g′(u)du

= res0

(g′(u)uk+1

)= coefficient of uk in g′(u).

(Integrating over a closed path around zero makes sense only for convergent powerseries with coefficients in C, however the result holds for all power series withcoefficients in R⊗Q.) Therefore,

g′(u) =∑

k>0

ϕ[CP k]uk.


Theorem E.2.5 then implies that g is the logarithm of the formal group law ϕ(FU ),and thus f is its exponential.

Now we can formulate the following criterion for the integrality of a genus:

Theorem E.3.4. Let R be a torsion-free ring and f(x) ∈ R ⊗ Q[[x]]. Thecorresponding genus ϕ : ΩU → R⊗Q takes values in R if and only if the coefficientsof the formal group law ϕ(FU )(u, v) = f(f−1(u) + f−1(v)) belong to R.

Proof. This follows from Theorem E.3.3 and Corollary E.2.4.

Example E.3.5. The universal genus maps a stably complex manifold M to itsbordism class [M ] ∈ ΩU and therefore corresponds to the identity homomorphismϕU : ΩU → ΩU . The corresponding characteristic class with coefficients in ΩU ⊗Q,the universal Hirzebruch genus, was studied in [47]; its corresponding series fU (x)is the exponential of the universal formal group law of geometric cobordisms.

Example E.3.6. We take R = Z in these examples.1. The top Chern number cn[M ] is a Hirzebruch genus, and its corresponding

f -series is f(x) = x1+x . The value of this genus on a stably complex manifold

(M, cT ) equals the Euler characteristic of M if cT is an almost complex structure.2. The L-genus L[M ] corresponds to the series f(x) = tanh(x) (the hyper-

bolic tangent). It is equal to the signature sign(M) by the classical Hirzebruchformula [160].

3. The Todd genus td[M ] corresponds to the series f(x) = 1 − e−x. Thecorresponding formal group law is given by F (u, v) = u + v − uv, compare Exam-ple (E.3). By Theorem E.3.4, the Todd genus is integral on any complex bordismclass. Furthermore, it takes value 1 on every complex projective space CP k.

The ‘trivial’ genus ε : ΩU → Z corresponding to the series f(x) = x gives riseto the augmentation transformation U∗ → H∗ from complex cobordism to ordinarycohomology (also known as the Thom homomorphism). More generally, for everygenus ϕ : ΩU → R and a space X we may set h∗ϕ(X) = U∗(X) ⊗ΩU

R. Undercertain conditions guaranteeing the exactness of the sequences of pairs (known asthe Landweber Exact Functor Theorem [191]) the functor h∗ϕ(·) gives rise to acomplex-oriented cohomology theory with the coefficient ring R.

As an example of this procedure, consider the genus corresponding to the 1-parameter extension of the multiplicative formal group law, see Example E.1.2. Itis also usually called the Todd genus, and takes values in the ring Z[β], deg β =−2. By interpreting β = 1 − η as the Bott element in the complex K-group

K0(S2) = K−2(pt) we obtain a homomorphism td : ΩU → K∗(pt). It gives riseto a multiplicative transformation U∗ → K∗ from complex cobordism to complexK-theory introduced by Conner and Floyd [82]. In this paper Conner and Floydproved that complex cobordism determines complex K-theory by means of theisomorphism K∗(X) = U∗(X) ⊗ΩU

Z[β], where the ΩU -module structure on Z[β]is given by the Todd genus. Their proof makes use of the Conner–Floyd Chernclasses; several proofs were given subsequently, including one which follows directlyfrom the Landweber exact functor theorem.


Example E.3.7. Another important example from the original work of Hirze-bruch is given by the χy-genus. It corresponds to the series

f(x) =1− e−x(1+y)1 + ye−x(1+y)

,

where y ∈ R is a parameter. Setting y = −1, y = 0 and y = 1 we get cn[M ], theTodd genus td[M ] and the L-genus L[M ] = sign(M) respectively.

If M is a complex manifold then the value χy[M ] can be calculated in terms ofthe Euler characteristics of Dolbeault complexes on M , see [160].

Exercises.

E.3.8. The formal group law corresponding to the χy-genus is given by

F (u, v) =u+ v + (y − 1)uv

1 + y · uv .

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Index

Action (of group), 420almost free, 169, 421effective, 421

free, 421proper, 200, 222

semifree, 300, 421transitive, 421

Acyclic (space), 269, 413

Affine equivalence, 8Alexander duality, 74, 118

Algebra, 387bigraded, 388connected, 387

exterior, 388finitely generated, 387

graded, 387graded-commutative, 387

free, 388

polynomial, 387multigraded, 388

with straightening law (ASL), 120Almost complex structure, 446

associated with omniorientation, 252

T -invariant, 252, 259, 457Ample divisor, 192

Annihilator (of module), 400Arrangement (of subspaces), 101

coordinate, 162Aspherical (space), 409, 242Associahedron, 40

generalised, 31Atiyah–Hirzebruch formula, 381

Axial function, 272, 310n-independent, 3112-independent, 310

Bar construction, 436Barnette sphere, 77, 79, 190

Barycentre (of simplex), 70Barycentric subdivision

of polytope, 71

of simplicial complex, 70of simplicial poset, 89, 128

Base (of Schlegel diagram), 77

Based map, 407Based space, 407Basepoint, 407

Basis (of free module), 388minimal, 392

Betti numbers (algebraic), 103, 124, 180Betti numbers (topological), 414

bigraded, 149

Bidegree, 388Bistellar equivalence, 85

Bistellar move, 84Blow-down, 317Blow-up

of T -graph, 316of T -manifold, 281

Boolean lattice, 87Bordism, 441

complex, 446

oriented, 445unoriented, 441

Borel construction, 422Borel spectral sequence, 233Bott manifold, 294

generalised, 306Bott–Taubes polytope, 42

Bott tower, 294generalised, 306

real, 308topologically trivial, 294

Boundary, 389

Bounded flag, 290Bounded flag manifold, 290, 295, 369

Bouquet (of spaces), 407Bruckner sphere, 81Buchsbaum complex, 116

Buchstaber invariant, 171Building set, 35

graphical, 40

Calabi–Eckmann manifold, 235Canonical line bundle, 424

Catalan number, 40Categorical quotient, 194

CAT(0) inequality, 72

483

484 INDEX

Cell complex, 408

Cellular approximation, 408Cellular chain, 415

Cellular map, 408

Chain complex, 389augmented, 412, 413

Chain homotopy, 389Characteristic function, 250

directed, 253

Characteristic matrix, 253refined, 254

Characteristic number, 452Characteristic pair, 251

combinatorial, 253

equivalence of, 252, 253Characteristic submanifold, 250, 267, 285

Charney–Davis Conjecture, 44

Chern–Dold character, 373Chow ring, 191

CHP (covering homotopy property), 409Classifying functor (algebraic), 436

Classifying space, 421

Clique, 71, 352Cobar construction, 343, 436

Cobordism, 443, 444complex, 447

equivariant

geometric, 365, 367homotopic, 365, 366

unoriented, 445

Coboundary, 389Cochain complex, 389

Cochain homotopy, 389Cocycle, 389

Cofibrant (object, replacement,approximation), 428

Cofibration, 411in model category, 428

Cofibre, 411Coformality, 349, 431, 436

Cohen–Macaulay

algebra, 400module, 400

simplicial complex, 112

simplicial poset, 127Cohomological rigidity, 307

Cohomologycellular, 416

of cochain complex, 389

simplicial, 412reduced, 413

singular, 414Cohomology product, 416

Cohomology product length, 161

Colimit, 179, 187, 322, 427Combinatorial equivalence

of polyhedral complexes, 76

of polytopes, 8

Combinatorial neighbourhood, 69

Complex orientable map, 447

Complete intersection algebra, 115Cone

over simplicial complex, 67

over space, 411

Cone (convex polyhedral), 62dual, 62

rational, 62

regular, 62

simplicial, 62

strongly convex, 62Connected sum

of manifolds, 351, 450

equivariant, 360

of simple polytopes, 12of simplicial complexes, 68

of stably complex manifolds, 451

Connection (on graph), 310

Contraction (of building set), 36Coordinate subspace, 162

Coproduct, 427

Core (of simplicial complex), 69

Cross-polytope, 10Cube, 8

standard, 8

topological, 89

Cubical complexabstract, 89

polyhedral, 90

topological, 89

Cubical subdivision, 94

Cup product, 416CW complex, 408

Cycle, 389

Cyclohedron, 42

Davis–Januszkiewicz space, 147, 324

Deformation retraction, 163

Degree (grading), 387external, 391

internal, 391

total, 391

Dehn–Sommerville relationsfor polytopes, 21, 25

for simplicial complexes, 117

for simplicial posets, 133

for triangulated manifolds, 134Depth (of module), 398

Diagonal approximation, 151

Diagonal map, 416

Diagram (functor), 427Diagram category, 427

Differential graded algebra (dg-algebra,dga), 390, 402

formal, 403

homologically connected, 402

minimal, 402

INDEX 485

simply connected, 402

Dimension

of module, 400

of polytope, 7of simplicial complex, 65

of simplicial poset, 87

Dolbeault cohomology, 230

Dolbeault complex, 230, 466

Double (simplicial), 172

Edge, 8

Eilenberg–Mac Lane space, 408

Eilenberg–Moore spectral sequence, 425

Elliptic genus, 372Equivariant bundle, 423

Equivariant characteristic class, 423

Equivariant cohomology, 423

of T -graph, 311

Equivariant map, 421

Euler class

in cobordism, 449in equivariant cobordism, 368

in equivariant cohomology, 267, 423

Euler formula, 21

Euler characteristic, 259, 383, 412, 413, 466

Eulerian poset, 133

Exact sequence, 389Excision, 415

Exponential (of formal group law), 460

Faceof cubical complex, 89

of cone, 62

of manifold with corners, 247, 269

of manifold with faces, 247

of polytope, 7

of simplicial complex, 65of simplicial poset, 88

of T -graph, 311

Face category (of simplicial complex), 88,322, 427

Face coalgebra, 342

Face poset, 8

Face ring (Stanley–Reisner ring)

of manifold with corners, 271of simple polytope, 98

of simplicial complex, 98

exterior, 148


Facet, 8, 247, 311

Face truncation, 11Fan, 62, 79

complete, 62

normal, 63

rational, 62

regular, 62

simplicial, 62

Fat wedge, 144

Fibrant (object, replacement,approximation), 428

Fibration, 409

in model category, 428locally trivial, 408

Fibre bundle, 408

associated (with G-space), 422

Fixed point, 421

Fixed point data, 376Flag complex (simplicial), 71, 347

Flagtope (flag polytope), 44

Folding map, 88

Formal group law, 459of geometric cobordisms, 461

linearisable, 459

universal, 460

Formality

in model category, 431integral, 324

of dg-algebra, 403

of space 175, 323, 327, 419

F -polynomial (of polytope), 20Frolicher spectral sequence, 233

Full subcomplex (of simplicial complex), 69

Fundamental group, 407

Fundamental homology class, 162f -vector (face vector)

of cubical complex, 90

of polytope, 20



Gal Conjecture, 72

Gale diagram, 17, 214, 242

combinatorial, 19, 213Gale duality, 16, 210, 216, 217, 232

Gale transform, 16

Ganea’s Theorem, 148

g-conjecture, 80, 117Generalised homology theory, 442

Generalised Lower Bound Conjecture, 28,194

Generating series

of face polynomials, 57

of polytopes, 55Genus, 372, 463

Geometric cobordism, 449

Geometric quotient, 195

Geometric realisationof cubical complex, 90


of simplicial set, 430

Ghost vertex, 65GKM-graph, 309

GKM-manifold, 309

Golod

ring, 178, 352

simplicial complex, 178

486 INDEX

Gorenstein, Gorenstein*

simplicial complex, 115, 159

simplicial poset, 131Goresky–MacPherson formula, 167

Graph

chordal, 352

of polytope, 14simple, 14, 40

Graph-associahedron, 40

Graph product, 348

g-theorem, 29, 193

g-vectorof polytope, 20


Gysin homomorphism

in equivariant cohomology, 267, 423in cobordism, 369, 449

Gysin–Thom isomorphism, 377, 452

Half-smash product (left, right), 329

Hamiltonian action, 201, 300

Hamiltonian-minimal submanifold, 237

Hamiltonian vector field, 236Hard Lefschetz Theorem, 30, 193

Hauptvermutung, 82

Heisenberg group, 420

HEP (homotopy extension property), 410Hermitian quadric, 208

Hilton–Milnor Theorem, 144

Hirzebruch genus, 372, 463

equivariant, 373fibre multiplicative, 374

rigid, 372, 374

universal, 466

Hirzebruch surface, 188

H-minimal submanifold, 237Hodge algebra, 120

Hodge number, 230

Homological dimension (of module), 391

Homologycellular, 416

of chain complex, 389

simplicial, 412

reduced, 412singular, 413

of pair, 414

reduced, 413

Homology polytope, 269Homology sphere, 82

Homotopy, 407

Homotopy category (of model category),429

Homotopy cofibre, 411

Homotopy colimit, 322, 438Homotopy equivalence, 407

Homotopy fibre, 410

Homotopy group, 407

Homotopy Lie algebra, 349, 417, 431

Homotopy limit, 322, 438

Homotopy quotient, 422Homotopy type, 407Hopf Conjecture, 72

Hopf equation, 56Hopf line bundle, 424

Hopf manifold, 227H-polynomial (of polytope), 20

hsop (homogeneous system of parameters),400

in face rings, 111, 121

h-vectorof polytope, 20

of simplicial complex, 66of simplicial poset, 121

Hyperplane cut, 11

Intersection homology, 30, 193Intersection poset (of arrangement), 167Isotropy representation, 422

Join (least common upper bound), 88, 118

Join (operation)of simplicial complexes, 67

of simplicial posets, 179of spaces, 327

Klein bottle, 239

Koszul algebra, 395Koszul complex, 395

Koszul resolution, 392

Lagrangian immersion, 236

Lagrangian submanifold, 236Landweber Exact Functor Theorem, 466

Latching functor, 432Lattice, 186, 215

Lefschetz pair, 139Left lifting property, 411

Leibniz identity, 390L-genus (signature), 466

Limit (of diagram), 427Link (in simplicial complex), 68, 128

Link (of intersection of quadrics), 213Localisation formula, 376

Locally standard (torus action), 246, 263Logarithm (of formal group law), 460Loop functor (algebraic), 436

Loop space, 338, 409Lower Bound Theorem, 27

lsop (linear system of parameters), 400in face rings, 111, 121

integral, 112LVM-manifold, 219, 229

Manifold with corners, 139, 247

face-acyclic, 269nice, 247

Manifold with faces, 247

INDEX 487

Mapping cone, 411Mapping cylinder, 411

Massey product, 175, 404indecomposable, 178indeterminacy of, 404trivial (vanishing), 404

Matching functor, 432

Mayer–Vietoris sequence, 415Maximal action (of torus), 229Meet (greatest common lower bound), 88,

118Milnor hypersurface, 454

Minimal modelof dg-algebra, 402, 431of space, 419

Minimal submanifold, 237Minkowski sum, 31

Missing face, 71, 340, 347Model (of dg-algebra), 402Model category, 428Module, 387

finitely generated, 387free, 388graded, 387projective, 388

Moment-angle complex, 137, 179

real, 140Moment-angle manifold, 140, 203

polytopal, 140, 212non-formal, 176

Moment map, 201, 215proper, 201, 216

Monoid, 142Monomial ideal, 98Moore loops, 338

Multi-fan, 245, 385Multidegree, 388Multigrading, 106, 154, 180M -vector, 29

Nerve complex, 66

Nested set, 36Nestohedron, 36Nilpotent space, 418Noether Normalisation Lemma, 400

Non-PL sphere, 80, 82Normal complex structure, 446

Omniorientation, 252, 267, 285, 315Opposite category, 427Orbifold, 189Orbit (of group action), 421

Orbit space, 421Order complex (of poset), 71Overcategory, 428

Path space, 409Pair (of spaces), 407

Perfect elimination order, 352

Permutahedron, 34

Picard group, 201PL map, 67

homeomorphism, 67

PL manifold, 82PL sphere, 76

Poincare algebra, 116Poincare–Atiyah duality, 448

Poincare duality, 415

Poincare duality space, 159Poincare series 391, 349

of face ring, 100Poincare sphere, 82

Pointed map, 407

Pointed space, 407Polar set, 9

Polyhedral complex, 76

Polyhedral product, 141, 179, 321Polyhedral smash product, 333

Polyhedron (convex), 7Delzant, 216

Polyhedron (simplicial complex), 65

Polytopal sphere, 79Polytope

combinatorial, 8convex, 7

cyclic, 13

Delzant, 63, 190, 204dual, 10

generic, 9

lattice, 189neighbourly, 13

nonrational, 190polar, 9

regular, 10

self-dual, 10simple, 9, 52

simplicial, 9, 52stacked, 27

triangle-free, 45

Polytope algebra, 30Pontryagin algebra, 339

Pontryagin product, 417

Pontryagin–Thom map, 423, 442Poset (partially ordered set), 8

Poset category, 427Positive orthant, 9, 136

Presentation (of a polytope byinequalities), 8

generic, 9irredundant, 8

rational, 216Primitive (lattice vector), 62

Principal bundle, 421

Principal minor (of matrix), 298Product

of building sets, 53

of polytopes, 11

488 INDEX

Product (categorical), 427Product (in cobordism), 448

Projective dimension (of module), 391Pseudomanifold, 157, 314

orientable, 158Pullback, 427Pushout, 427

Quadratic algebra, 98, 347

Quasi-isomorphism, 390, 402, 430Quasitoric manifold, 250, 326, 382

almost complex structure on, 259canonical smooth structure on, 255

complex structure on, 259equivalence of, 252, 253

Quillen pair (of functors), 429Quotient space, 421

Rank function, 87Rational equivalence, 418

Rational homotopy type, 418Ray’s basis, 371Redundant inequality, 8Reedy category, 432, 438

Regular sequence, 397Regular subdivision, 127Regular value (of moment map), 201, 216Reisner Theorem, 114Resolution (of module)

free, 391minimal, 392projective, 391

Restriction (of a building set), 32Restriction map

algebraic, 100, 111, 120, 272in equivariant cohomology, 265

Right-angled Artin group, 348Right-angled Coxeter group, 348

Right lifting property, 411Ring of polytopes, 50

Samelson product, 339, 417higher, 340, 431

Schlegel diagram, 76Segre embedding, 454

Seifert fibration, 230Sign (of fixed point), 257, 382, 457Signature, 383, 466Simplex, 8, 88

abstract, 65regular, 8standard, 8

Simplicial cell complex, 88Simplicial chain, 411

Simplicial cochain, 412Simplicial complex

abstract, 65Alexander dual, 73

geometric, 65

pure, 65

shifted, 178, 332

Simplicial manifold, 82Simplicial map, 66

isomorphism, 66

nondegenerate, 66

Simplicial object (in category), 427Simplicial poset, 87

dual (of manifold with corners), 270

Simplicial set, 427

Simplicial sphere, 76

Simplicial subdivistion, 67Simplicial wedge, 172

Singular chain, 413

of pair, 414

Singular cochain, 414Skeleton (of cell complex), 408

Slice Theorem, 422

Small category, 427

Small cover, 239, 289, 308Smash product, 329, 407

Stabiliser (of group action), 421

Stably complex structure, 256, 445

T -invariant, 457Stanley–Reisner ideal, 98

Star, 68, 128, 224

Starshaped sphere, 79, 226

Stasheff polytope, 31Stationary subgroup, 421

Steinitz Problem, 81

Steinitz Theorem, 77

Stellahedron, 42

Stellar subdivisionof simplicial complex, 84


Straightening relation, 120

Subcomplex of simplicial complex, 65Sullivan algebra (of piecewise polynomial

differential forms), 418

Supporting hyperplane, 7

Suspension

of module, 388of simplicial complex, 67

of space, 411

Suspension isomorphism, 415

Symplectic manifold, 201

Symplectic reduction, 201, 215, 242Symplectic quotient, 202, 215

Sysygy, 391

Tangential representation, 422

Tautological line bundle, 424

Tensor product (of modules), 388

T -graph (torus graph), 311Thom class, 273, 312, 423

Thom space, 442

T -manifold, 245

Todd genus, 384, 466

INDEX 489

Topological fan, 286complete, 286

Topological monoid, 430Topological toric manifold, 284

Tor-algebra, 103Toral rank, 169Toric manifold, 191, 242, 248

Hamiltonian, 204, 218over cube, 294projective, 191

Toric space, 2, 136Toric variety, 30, 185

affine, 186

projective, 189real, 289, 239

Torusalgebraic, 185standard, 136

Torus manifold, 263Triangulated manifold, 82Triangulated sphere, 27, 76Triangulation, 61, 67

neighbourly, 81

Tropical geometry, 289Truncated cube, 46Truncated simplex, 44

Undercategory, 428Underlying simplicial complex (of fan), 67Unipotent (upper triangular matrix), 298Universal toric genus, 366, 368Upper Bound Theorem, 26

Vector bundle, 422Vertex

of polytope, 8of simplicial complex, 65of simplicial poset, 87

Vertex truncation, 11Volume polynomial, 30

Weak equivalenceof dg-algebras, 402in model category, 428

Wedge (of spaces), 144, 407of spheres, 178, 328, 352

Weight (of torus representation), 256, 382,457

Weight graph, 308Weight lattice (of torus), 215

Whitehead product, 417higher, 340

Yoneda algebra, 436

Zonotope, 35

γ-polynomial, 23γ-vector

of polytope, 23

of triangulated sphere, 72χa,b-genus, 380χy-genus, 466

2-truncated cube, 4624-cell, 105-lemma, 390

toric topology - arxiv.org e-print archivetoric topology victor m. buchstaber taras e. panov steklov...

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